Received 25 November 2015; accepted 25 January 2016; published 29 January 2016
P 1.1. In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index.
Originally, such transfer mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin’s reciprocity isomorphism ( , §4, Allgemeines Reziprozitätsgesetz, p. 361) to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups ( , §2, p. 50).
However, independently of number theoretic applications, a natural partial order on the kernels and targets of Artin transfers, has recently been found to be compatible with parent-child relations between finite p-groups, where p denotes a prime number. Such ancestor-descendant relations can be visualized conveniently in des- cendant trees ( , §4, pp. 163-164).
Consequently, Artin transfers provide valuable information for classifying finite p-groups by kernel-target patterns and for searching and identifying particular groups in descendant trees by looking for patterns defined by kernels and targets of Artin transfers. These strategies of pattern recognition are useful not only in purely group theoretic context, but also, most importantly, for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers. The reason is that the unramified extensions of a base field contain information in the shape of capitulation patterns and class group structures, and these arithmetic invariants can be translated into group theoretic data on transfer kernels and targets by means of Artin’s reciprocity law of class field theory. The natural partial order on Artin patterns admits termination criteria for a search through a descendant tree with the aid of recursive executions of the p-group generation algorithm by Newman  and O’Brien  .
P 1.2. The organization of this article is as follows. The detailed theory of the transfer will be developed in §§ 2 and 3, followed by computational implementations in § 4. It is our intention to present more than the least common multiple of the original papers by Schur  and Artin  and the relevant sections of the text books by Hall  , Huppert  , Gorenstein  , Aschbacher  , Doerk and Hawkes  , Smith and Tabachnikova  , and Isaacs  .
However, we shall not touch upon fusion and focal subgroups, which form the primary goal of the mentioned authors, except Artin. Our focus will rather be on a sound foundation of Artin patterns, consisting of families of transfer kernels and targets, and their stabilization, resp. polarization, in descendant trees of finite p-groups. These phenomena arise from a natural partial order on Artin patterns which is compatible with ancestor- descendant relations in trees, and is established in its most general form in §§5 and 6.
Since our endeavour is to give the most general view of each partial result, we came to the conviction that categories, functors and natural transformations are the adequate tools for expressing the appropriate range of validity for the facts connected with the partial order relation on Artin patterns. Inspired by Bourbaki’s method of exposition  , Appendix on induced homomorphisms, which is separated to avoid a disruption of the flow of exposition, goes down to the origins exploiting set theoretic facts concerning direct images and inverse pre-images of mappings which are crucial for explaining the natural partial order of Artin patterns.
2. Transversals and Their Permutations
2.1. Transversals of a Subgroup
Let G be a group and be a subgroup of finite index.
Definition 2.1. See also ( , p. 1013); ( , (1.5.1), p. 11); ( , Satz 2.5, p. 5).
1). A left transversal of H in G is an ordered system of representatives for the left cosets of H in
G such that is a disjoint union.
2). Similarly, a right transversal of H in G is an ordered system of representatives for the right
cosets of H in G such that is a disjoint union.
Remark 2.1. For any transversal of H in G, there exists a unique subscript such that, resp.. The element, resp., which represents the principal coset (i.e., the subgroup H itself) may be replaced by the neutral element 1.
Lemma 2.1. See also ( , p. 1015); ( , (1.5.2), p. 11); ( , Satz 2.6, p. 6).
1). If G is non-abelian and H is not a normal subgroup of G, then we can only say that the inverse elements of a left transversal form a right transversal of H in G.
2). However, if is a normal subgroup of G, then any left transversal is also a right transversal of H in G.
Proof. 1). Since the mapping, is an involution, that is a bijection which is its own inverse, we see that
2). For a normal subgroup, we have for each.
Let be a group homomorphism and be a left transversal of a subgroup H in G with finite index. We must check whether the image of this transversal under the homomorphism is a transversal again.
Proposition 2.1. The following two conditions are equivalent.
1). is a left transversal of the subgroup in the image with finite
We emphasize this important equivalence in a formula:
Proof. By assumption, we have the disjoint left coset decomposition which comprises two
Firstly, the group is a union of cosets,
and secondly, any two distinct cosets have an empty intersection, for.
Due to the properties of the set mapping associated with, the homomorphism maps the union to another union
but weakens the equality for the intersection to a trivial inclusion
To show that the images of the cosets remain disjoint we need the property of the homo- morphism.
Suppose that for some,
then we have for certain elements.
Multiplying by from the left and by from the right, we obtain
Since, this implies, resp., and thus. (This part of the proof is also covered by ( , Thm. X. 21, p. 340) and, in the context of normal subgroups instead of homomorphisms, by ( , Thm. 2.3.4, p. 29) and ( , Satz 3.10, p. 16))
Conversely, we use contraposition.
If the kernel of is not contained in the subgroup H, then there exists an element such that.
But then the homomorphism maps the disjoint cosets
to equal cosets.
2.2. Permutation Representation
P 2.1. Suppose is a left transversal of a subgroup of finite index in a group G. A fixed element gives rise to a unique permutation of the left cosets of H in G by left multiplication such that
Similarly, if is a right transversal of H in G, then a fixed element gives rise to a unique permutation of the right cosets of H in G by right multiplication such that
The elements, resp., , of the subgroup H are called the monomials associated with x with respect to, resp..
Definition 2.2 See also ( , Hauptsatz 6.2, p. 28).
The mapping, resp., is called the permutation representation of G in with respect to, resp..
Lemma 2.2. For the special right transversal associated to the left transversal, we have the following relations between the monomials and permutations corresponding to an element:
Proof. For the right transversal, we have, for each.
On the other hand, for the left transversal, we have
, for each.
This relation simultaneously shows that, for any, the permutation representations and the associated monomials are connected by
for each. □
3. Artin Transfer
Let G be a group and be a subgroup of finite index. Assume that, resp., is a left, resp. right, transversal of H in G with associated permutation representation, , resp., such that, resp., for.
Definition 3.3. See also ( , p. 1014); ( , §2, p. 50); ( , (14.2.2-4), p. 202); ( , p. 413); ( , p. 248); ( , p. 197); ( , Dfn.(17.1), p. 60); ( , p. 154); ( , p. 149); ( , p. 2).
The Artin transfer from G to the abelianization of H with respect to, resp., is defined by
Remark 3.1. I.M. Isaacs  , p. 149 calls the mapping, , resp.,
the pre-transfer from G to H. The pre-transfer can be composed with a homomorphism from H into
an abelian group A to define a more general version of the transfer, , resp.
, from G to A via, which occurs in the book by D. Gorenstein ( , p. 248). Taking the
natural epimorphism, , yields the Definition 3.3 of the Artin transfer in its original form by I. Schur ( , p. 1014) and by E. Artin ( , §2, p. 50), which has also been dubbed Verlagerung by H. Hasse ( , §27.4, pp. 170-171). Note that, in general, the pre-transfer is neither independent of the transversal nor a group homomorphism.
3.1. Independence of the Transversal
Assume that is another left transversal of H in G such that.
Proposition 3.1. See also ( , p. 1014); ( , Thm. 14.2.1, p. 202); ( , Hilfssatz 1.5, p. 414); ( , Thm. 3.2, p. 246); ( , (37.1), p.198); ( , Thm.(17.2), p.61); ( , p.154); ( , Thm.5.1, p.149); ( , Prop.2, p. 2).
The Artin transfers with respect to (g) and coincide,.
Proof. There exists a unique permutation such that, for all. Consequently, , resp. with, for all. For a fixed element, there exists a unique permutation such that we have
for all. Therefore, the permutation representation of G with respect to is given by, resp., for. Furthermore, for the connection between the elements
and, we obtain
for all. Finally, due to the commutativity of the quotient group and the fact that and are permutations, the Artin transfer turns out to be independent of the left transversal:
as prescribed in Definition 3.1, Equation (3.1). □
It is clear that a similar proof shows that the Artin transfer is independent of the choice between two different right transversals. It remains to show that the Artin transfer with respect to a right transversal coincides with the Artin transfer with respect to a left transversal.
For this purpose, we select the special right transversal associated to the left transversal, as explained in Lemma 2.1 and Lemma 2.2.
Proposition 3.2. The Artin transfers with respect to and coincide,.
Proof. Using (2.4) in Lemma 2.2 and the commutativity of, we consider the expression
The last step is justified by the fact that the Artin transfer is a homomorphism. This will be shown in the following subsection 3.2. □
3.2. Artin Transfers as Homomorphisms
Let be a left transversal of H in G.
Theorem 3.1. See also ( , p. 1014); ( , Thm. 14.2.1, p. 202); ( , Hauptsatz 1.4, p. 413); ( , Thm. 3.2, p. 246); ( , (37.2), p.198); ( , Thm.(17.2), p.61); ( , p. 155); ( , Thm.5.2, p. 150); ( , Prop.1, p. 2).
The Artin transfer, and the permutation representation
are group homomorphisms:
Proof. Let be two elements with transfer images and
. Since is abelian and is a permutation, we can change the order of the factors in the following product:
This relation simultaneously shows that the Artin transfer and the permutation representation are homomorphisms, since and, in a covariant way. □
3.3. Monomial Representation
Let, resp., be a left, resp. right, transversal of a subgroup H in a group G. Using the monomials, resp., associated with an element according to Equation (2.2), resp. (2.3), we define the following maps.
Definition 3.2. The mapping, respectively , is called the monomial representation of G in with respect to, resp..
P 3.1. It is illuminating to restate the homomorphism property of the Artin transfer in terms of the monomial
representation. The images of the factors are given by and
. In the proof of Theorem 3.1, the image of the product turned out to be, which is a very peculiar law of com- position discussed in more detail in the sequel.
The law reminds of the crossed homomorphisms in the first cohomology group of a G-module M, which have the property, for.
These peculiar structures can also be interpreted by endowing the cartesian product with a special law of composition known as the wreath product of the groups H and with respect to the set.
Definition 3.3. For, the wreath product of the associated monomials and permutations is given by
Theorem 3.2. See also ( , Thm.14.1, p. 200); ( , Hauptsatz 1.4, p. 413).
This law of composition on causes the monomial representation
also to be a homomorphism. In fact, it is a faithful representation, that is an injective homomorphism, also called a monomorphism or embedding, in contrast to the permutation representation.
Proof. The homomorphism property has been shown above already. For a homomorphism to be injective, it suffices to show the triviality of its kernel. The neutral element of the group endowed with the wreath product is given by, where the last 1 means the identity permutation. If , for some xÎG, then and consequently, for all. Finally, an application of the inverse inner automorphism with yields, as required for injectivity.
The permutation representation cannot be injective if G is infinite or at least of an order bigger than, the factorial of n. □
Remark 3.2. Formula (3.4) is an example for the left-sided variant of the wreath product on. However, we point out that the wreath product with respect to a right transversal of H in G appears in its right-sided variant
which implies that the permutation representation, is a homomorphism with respect to the opposite law of composition on, in a contravariant manner.
It can be shown that the left-sided and the right-sided variant of the wreath product lead to isomorphic group structures on.
A related viewpoint is taken by M. Hall ( , p. 200), who uses the multiplication of monomial matrices to describe the wreath product. Such a matrix can be represented in the form as the product of an invertible diagonal matrix over the group ring K[H], where K denotes a field, and the permutation matrix associated with the permutation. Multiplying two such monomial matrices yields a law of composition identical to the wreath product in the right-sided variant,
Whereas B. Huppert ( , p. 413) uses the monomial representation for defining the Artin transfer by composition with the unsigned determinant, we prefer giving the immediate Definition 3.3 and merely illustrating the homomorphism property of the Artin transfer with the aid of the monomial representation.
3.4. Composition of Artin Transfers
Let G be a group with nested subgroups such that the indices, and are finite.
Theorem 3.3. See also ( , Thm.14.2.1, p. 202); ( , Satz 1.6, p. 415); ( , Lem.(17.3), p. 61); ( , Thm.10.8, p. 301); ( , Prop.3, p. 3).
Then the Artin transfer is the compositum of the induced transfer (in the sense of Corollary 7.1 or Corollary 7.3 in the Appendix) and the Artin transfer, i.e.,
This can be seen in the following manner.
Proof. If is a left transversal of H in G and is a left transversal of K in H, that is
and, then is a disjoint left coset decomposition of G with
respect to K. See also ( , Thm.1.5.3, p. 12); ( , Satz 2.6, p. 6). Given two elements and, there exist unique permutations, and, such that the associated monomials are given by
, for each, and, for each.
Then, using Corollary 7.3, we have
For each pair of subscripts and, we put and obtain
resp.. Thus, the image of x under the Artin transfer is given by
3.5. Wreath Product of Sm and Sn
P 3.2. Motivated by the proof of Theorem 3.3, we want to emphasize the structural peculiarity of the monomial representation
which corresponds to the compositum of Artin transfers, defining
for a permutation, and using the symbolic notation for all pairs of subscripts,.
The preceding proof has shown that. Therefore, the action of the permutation on the set is given by. The action on the second component j depends
on the first component i (via the permutation), whereas the action on the first component i is independent of the second component j. Therefore, the permutation can be identified with the multiplet, which will be written in twisted form in the sequel.
The permutations, which arise as second components of the monomial representation
are of a very special kind. They belong to the stabilizer of the natural equipartition of the set [1, n] × [1, m] into the n rows of the corresponding matrix (rectangular array). Using the peculiarities of the composition of Artin transfers in the previous section, we show that this stabilizer is isomorphic to the wreath product of the symmetric groups and with respect to, whose underlying set is endowed with the following law of composition in the left-sided variant.
This law reminds of the chain rule for the Fréchet derivative in
xÎE of the compositum of differentiable functions and between complete normed spaces.
The above considerations establish a third representation, the stabilizer representation,
of the group G in the wreath product, similar to the permutation representation and the monomial representation. As opposed to the latter, the stabilizer representation cannot be injective, in general. For instance, certainly not, if G is infinite.
Formula (3.7) proves the following statement.
Theorem 3.4. The stabilizer representation of the group G in
the wreath product of symmetric groups is a group homomorphism.
3.6. Cycle Decomposition
Let be a left transversal of a subgroup of finite index in a group G. Suppose the element gives rise to the permutation of the left cosets of H in G such that
, resp., for each.
Theorem 3.5. See also ( , §2, p. 50); ( , §27.4, p. 170); ( , Hilfssatz 1.7, p. 415); ( , Thm.3.3, p. 249); ( , (37.3), p. 198); ( , p. 154); ( , Lem.5.5, p. 153); ( , p. 5).
If the permutation has the decomposition into pairwise disjoint (and thus commuting)
cycles of lengths, which is unique up to the ordering of the cycles, more explicitly, if
for, and, then the image of under the Artin transfer is given by
Proof. The reason for this fact is that we obtain another left transversal of H in G by putting for and, since
is a disjoint decomposition of G into left cosets of H.
Let us fix a value of. For, we have
However, for, we obtain
P 3.3. The cycle decomposition corresponds to a double coset decomposition of the group
G modulo the cyclic group and modulo the subgroup H. It was actually this cycle decomposition form of the transfer homomorphism which was given by E. Artin in his original 1929 paper ( , §2, p. 50).
3.7. Transfer to a Normal Subgroup
P 3.4. Now let be a normal subgroup of finite index in a group G. Then we have, for all, and there exists the quotient group of order n. For an element, we let denote the order of the coset in, and we let be a left transversal of the subgroup in G, where.
Theorem 3.6. See also ( , §27.4, VII, p. 171).
Then the image of under the Artin transfer is given by
Proof. is a cyclic subgroup of order f in, and a left transversal of the subgroup
in G, where and is the corresponding disjoint left coset decomposition,
can be refined to a left transversal with disjoint left coset decomposition
of H in G. Hence, the formula for the image of x under the Artin transfer in the previous section takes the particular shape
with exponent f independent of j. □
Corollary 3.1. See also ( , Lem.10.6, p. 300) for a special case.
In particular, the inner transfer of an element is given as a symbolic power
with the trace element
of H in G as symbolic exponent.
The other extreme is the outer transfer of an element which generates G modulo H, that is. It is simply an nth power
Proof. The inner transfer of an element, whose coset is the principal set in of order, is given as the symbolic power
with the trace element
of H in G as symbolic exponent.
The outer transfer of an element which generates G modulo H, that is, whose coset is generator of with order, is given as the nth power
P 3.5. Transfers to normal subgroups will be the most important cases in the sequel, since the central concept of this article, the Artin pattern, which endows descendant trees with additional structure, consists of targets and kernels (§5) of Artin transfers from a group G to intermediate groups between G and its com- mutator subgroup. For these intermediate groups we have the following lemma.
Lemma 3.1. All subgroups of a group G which contain the commutator subgroup are normal subgroups.
Proof. Let. If H were not a normal subgroup of G, then we had for some element. This would imply the existence of elements and such that, and consequently the commutator would be an element in in contradiction to. □
Explicit implementations of Artin transfers in the simplest situations are presented in the following section.
4. Computational Implementation
4.1. Abelianization of Type (p, p)
P 4.1. Let G be a pro-p group with abelianization of elementary abelian type. Then G has maximal subgroups of index. In this particular case, the Frattini
subgroup, which is defined as the intersection of all maximal subgroups, coincides with the
commutator subgroup, since the latter contains all pth powers, and thus we have.
For each, let be the Artin transfer homomorphism from G to the abelianization of. According to Burnside's basis theorem, the group G has generator rank and can therefore be generated as by two elements such that. For each of the normal subgroups
, we need a generator with respect to, and a generator of a transversal
such that and.
A convenient selection is given by
Then, for each, it is possible to implement the inner transfer by
according to Equation (3.13) of Corollary 3.1, which can also be expressed by a product of two pth powers,
and to implement the outer transfer as a complete pth power by
according to Equation (3.15) of Corollary 3.1. The reason is that and in the quotient group.
It should be pointed out that the complete specification of the Artin transfers also requires explicit knowledge of the derived subgroups. Since is a normal subgroup of index p in, a certain general reduction is possible by ( , Lem.2.1, p. 52), but an explicit pro-p pre- sentation of G must be known for determining generators of, whence
4.2. Abelianization of Type (p2, p)
P 4.2. Let G be a pro-p group with abelianization of non-elementary abelian type. Then G has maximal subgroups of index and subgroups
Figure 1 visualizes this smallest non-trivial example of a multi-layered abelianization ( , Dfn.3.1- 3, p. 288).
For each, let, resp., be the Artin transfer homo- morphism from G to the abelianization of, resp.. Burnside’s basis theorem asserts that the group G has generator rank and can therefore be generated as by two elements such that
Figure 1. Layers of subgroups for.
We begin by considering the first layer of subgroups. For each of the normal subgroups , we select a generator
These are the cases where the factor group is cyclic of order. However, for the distinguished maximal subgroup, for which the factor group is bicyclic of type, we need two generators
Further, a generator of a transversal must be given such that, for each. It is convenient to define
Then, for each, we have the inner transfer
which equals, and the outer transfer
Now we continue by considering the second layer of subgroups. For each of the normal subgroups , we select a generator
such that. Among these subgroups, the Frattini subgroup is par- ticularly distinguished. A uniform way of defining generators of a transversal such that, is to set
Since, but on the other hand and, for, with the single exception that, we obtain the following expressions for the inner transfer
and for the outer transfer
for. Again, it should be emphasized that the structure of the derived subgroups and must be known explicitly to specify the action of the Artin transfers completely.
5. Transfer Kernels and Targets
P 5.1. After our thorough treatment of the general theory of Artin transfers in §§2 and 3, and their computational implementation for some simple cases in §4, we are now in the position to introduce Artin transfer patterns, which form the central concept of this article. They provide an incredibly powerful tool for classifying finite and infinite pro-p groups and for identifying a finite p-group G with sufficiently many assigned components of its Artin pattern by the strategy of pattern recognition. This is done in a search through the descendant tree with root by means of recursive applications of the p-group generation algorithm by Newman  and O’Brien  .
An Artin transfer pattern consists of two families of transfer targets, resp. kernels, which are also called multiplets, whereas their individual components are referred to as singulets.
5.1. Singulets of Transfer Targets
Theorem 5.1. Let G and T be groups. Suppose that is the image of G under a homomorphism, and is the image of an arbitrary subgroup. Then the following claims hold without any further necessary assumptions.
1) The commutator subgroup of V is the image of the commutator subgroup of U, that is
2) The restriction is an epimorphism which induces a unique epimorphism
Thus, the abelianization of V,
is an epimorphic image of the abelianization of U, namely the quotient of by the kernel of, which is given by
3) Moreover, the map is an isomorphism, and the quotients are isomorphic, if and only if
See Figure 2 for a visualization of this situation.
Proof. The statements can be seen in the following manner. The image of the commutator subgroup is given by
The homomorphism can be restricted to an epimorphism. According to Theorem 7.1, in particular, by the Formulas (7.5) and (7.4) in the appendix, the condition implies the existence of a uniquely determined epimorphism such that. The Isomor- phism Theorem in Formula (7.7) in the appendix shows that. Furthermore, by the Formulas (7.4) and (7.1), the kernel of is given explicitly by
Thus, is an isomorphism if and only if. □
P 5.2. Functor of derived quotients. In analogy to section §7.6 in the appendix, a covariant functor can be used to map a morphism of one category to an induced morphism of another category.
In the present situation, we denote by the category of groups and we define the domain of the functor F as the following category. The objects of the category are pairs consisting of a group G and a subgroup,
For two objects, the set of morphisms consists of epimor- phisms such that and, briefly written as arrows,
The functor from this category to the category of abelian groups maps a pair to the commutator quotient group of the subgroup, and
it maps a morphism to the induced epimorphism of the restriction,
Existence and uniqueness of have been proved in Theorem 5.1 under the assumption that, which is satisfied according to the definition of the arrow and automatically implies.
Figure 2. Induced homomorphism of derived quotients.
Definition 5.1. Due to the results in Theorem 5.1, it makes sense to define a pre-order of transfer targets on the image of the functor F in the object class of the category of abelian groups in the following manner.
For two objects, a morphism, and the images
let (non-strict) precedence be defined by
and let equality be defined by
if the induced epimorphism is an isomorphism.
Corollary 5.1. If both components of the pairs are restricted to Hopfian groups, then the pre-order of transfer targets is actually a partial order.
Proof. We use the functorial properties of the functor F. The reflexivity of the partial order follows from the functorial identity in Formula (7.14), and the transitivity is a consequence of the functorial compositum in Formula (7.15), given in the appendix. The antisymmetry might be a problem for infinite groups, since it is known that there exist so-called non-Hopfian groups. However, for finite groups, and more generally for Hop- fian groups, it is due to the implication . □
5.2. Singulets of Transfer Kernels
Suppose that G and T are groups, is the image of G under a homomorphism, and is the image of a subgroup of finite index. Let be the Artin transfer from G to.
Theorem 5.2. If, then the image of a left transversal of U in G is a left transversal of V in H, the index remains the same and is therefore finite, and the Artin transfer from H to exists.
1) The following connections exist between the two Artin transfers: the required condition for the composita of mappings in the commutative diagram in Figure 3,
and, consequently, the inclusion of the kernels,
2) A sufficient (but not necessary) condition for the equality of the kernels is given by
Figure 3. Epimorphism and Artin transfer.
See Figure 3 for a visualization of this scenario.
Proof. The truth of these statements can be justified in the following way. The first part has been proved in
Proposition 2.1 already: Let be a left transversal of U in G. Then is a disjoint union but the union is not necessarily disjoint. For, we have
element. However, if the condition is satisfied, then we are able to conclude that
, and thus.
Let be the epimorphism obtained in the manner indicated in the proof of Theorem 5.1 and Formula (5.2). For the image of under the Artin transfer, we obtain
Since, the right hand side equals, provided that is a left transversal of V in H, which is correct when. This shows that the diagram in Figure 3 is commutative, that is,. It also yields the connection between the permutations and the monomials for all. As a consequence, we obtain the inclusion
, if. Finally, if, then the previous section has shown that
is an isomorphism. Using the inverse isomorphism, we get, which proves the equation
. More explicitly, we have the following chain of equivalences and implications:
Conversely, only implies. Therefore, we cer-
tainly have if, which is, however, not necessary. □
P 5.3. Artin transfers as natural transformations. Artin transfers can be viewed as components of a natural transformation T between two functors and F from the following category to the usual category of groups.
The objects of the category are pairs consisting of a group G and a subgroup of finite index,
For two objects, the set of morphisms consists of epimor- phisms satisfying, , and the additional condition for their kernels, briefly written as arrows,
The forgetful functor from this category to the category of groups maps a pair. to its first component, and it maps a morphism to the underlying epimorphism.
The functor from to the category of groups maps a pair to the commutator quotient group of the subgroup U, and it maps a morphism to the induced epimorphism of the restriction. Note that we must abstain here from letting F map into the subcategory of abelian groups.
The system T of all Artin transfers fulfils the requirements for a natural transformation between these two functors, since we have
for every morphism of the category.
Definition 5.2. Due to the results in Theorem 5.2, it makes sense to define a pre-order of transfer kernels on the kernels of the components of the natural transformation T in the object class of the category of groups in the following manner.
For two objects, a morphism, and the images
let (non-strict) precedence be defined by
and let equality be defined by
if the induced epimorphism is an isomorphism.
Corollary 5.2. If both components of the pairs are restricted to Hopfian groups,
then the pre-order of transfer kernels is actually a partial order.
Proof. Similarly as in the proof of Corollary 5.1, we use the properties of the functor F. The reflexivity is due to the functorial identity in Formula (7.14). The transitivity is due to the functorial compositum in Formula (7.15), where we have to observe the relations, , and Formula (7.1) in the appendix for verifying the kernel relation
additionally to the image relation
The antisymmetry is certainly satisfied for finite groups, and more generally for Hopfian groups. □
5.3. Multiplets of Transfer Targets and Kernels
Instead of viewing various pairs which share the same first component G as distinct objects in the categories, resp., which we used for describing singulets of transfer targets, resp. kernels, we now consider a collective accumulation of singulets in multiplets. For this purpose, we shall define a new category of families, which generalizes the category, rather than the category. However, we have to pre- pare this definition with a criterion for the compatibility of a system of subgroups with its image under a homo- morphism.
Proposition 5.1. See also ( , Thm.2.3.4, p. 29); ( , Satz 3.10, p. 16); ( , Thm.2.4, p. 6); ( , Thm.X.21, p. 340).
For an epimorphism of groups, the associated set mappings
are inverse bijections between the following systems of subgroups
Proof. The fourth and fifth statement of Lemma 7.1 in the appendix show that usually the associated set mappings and of a homomorphism are not inverse bijections between systems of sub- groups of G and H. However, if we replace the homomorphism by an epimorphism with, then the Formula (7.2) yields the first desired equality
Guided by the property of all pre-images of, we define a re- stricted system of subgroups of the domain G,
and, according to Formula (7.1.), we consequently obtain the second required equality
which yields the crucial pair of inverse set bijections
P 5.4. After this preparation, we are able to specify the new category. The objects of the category
are pairs consisting of a group G and the family of all subgroups with finite index
where I denotes a suitable indexing set. Note that G itself is one of the subgroups.
The morphisms of the new category are subject to more restrictive conditions, which concern entire families of subgroups instead of just a single subgroup.
For two objects, the set of
morphisms consists of epimorphisms satisfying, the image conditions, and the kernel conditions, which imply the pre-image conditions, for all, briefly written as arrows,
Note that, in view of Proposition 5.1, we can always use the same indexing set I for the domain and for the codomain of morphisms, provided they satisfy the required kernel condition.
Now we come to the essential definition of Artin transfer patterns.
Definition 5.3. Let be an object of the category.
The transfer target type (TTT) of G is the family
The transfer kernel type (TKT) of G is the family
The complete Artin pattern of G is the pair
P 5.5. The natural partial order on TTTs and TKTs is reduced to the partial order on the components, according to the Definitions 5.1 and 5.2.
Definition 5.4. Let be two objects of the category, where all members of the families and are Hopfian groups.
Then (non-strict) precedence of TTTs is defined by
and equality of TTTs is defined by
(Non-strict) precedence of TKTs is defined by
and equality of TKTs is defined by
We partition the indexing set I in two disjoint components, according to whether components of the Artin pattern remain fixed or change under an epimorphism.
Definition 5.5. Let be two objects of the category, and let be a morphism between these objects.
The stable part and the polarized part of the Artin pattern of G with respect to are defined by
Accordingly, we have
Note that the precedence of polarized targets is strict as opposed to polarized kernels.
5.4. The Artin Pattern on a Descendant Tree
Firstly, a basic relation between parent and child (also called immediate descendant), corre- sponding to a directed edge of the tree, for any vertex which is different from the root R of the tree.
Secondly, an induced non-strict partial order relation, for some integer, between ancestor and descendant, corresponding to a path of directed edges, for an arbitrary vertex, that is, the ancestor is an iterated parent of the descendant. Note that only an empty path with starts from the root R of the tree, which has no parent.
Just a brief justification of the partial order: Reflexivity is due to the relation. Transitivity
follows from the rule. Antisymmetry is a consequence of the absence of cycles, that is, implies and thus.
P 5.7. The category of a tree. Now let be a rooted directed tree whose vertices are groups. Then we define, the category associated with, as a subcategory of the category which was introduced in the Formulas (5.23) and (5.24).
The objects of the category are those pairs in the object class of the category whose
first component is a vertex of the tree,
The morphisms of the category are selected along the paths of the tree only.
For two objects, the set of morphisms is either empty or consists of a single element only,
In the case of an ancestor-descendant relation between H and G, the specification of the supercategory
enforces the following constraints on the unique morphism: the image relations and the kernel relations, for all.
P 5.8. At this position, we must start to be more concrete. In the descendant tree of a group R, which is the root of the tree, the formal parent operator gets a second meaning as a natural projection, , from the child G onto its parent, which is always the quotient of G by a suitable normal subgroup. To be precise, the epimorphism with kernel is actually dependent on its domain G. Therefore, the formal power is only a convenient
abbreviation for the compositum.
As described in  , there are several possible selections of the normal subgroup N in the parent definition. Here, we would like to emphasize the following three choices of characteristic subgroups N of the child G. If p denotes a prime number and is the descendant tree of a finite p-group R, then it is usual to take for
1) either the last non-trivial member of the lower central series of G
2) or the last non-trivial member of the lower exponent-p central series of G
3) or the last non-trivial member of the derived series of G,
where denotes the nilpotency class, the lower exponent p-class, and the derived length of G, respectively.
Note that every descendant tree of finite p-groups is subtree of a descendant tree with abelian root. Therefore, it is no loss of generality to restrict our attention to descendant trees with abelian roots.
Theorem 5.3. A uniform warranty for the comparability of the Artin patterns of all vertices G of a descendant tree of finite p-groups with abelian root R, in the sense of the natural partial
order, is given by the following restriction of the family of subgroups in the corresponding object of the category. The restriction depends on the definition of a parent in the
1) for all, when with.
2) for all, when with.
3) for all, when with.
Proof. If parents are defined by with, then we have and for any. The largest of these kernels arises for. Therefore, uniform comparability of Artin patterns is warranted by the restriction for all.
The parent definition with implies and for any. The largest of these kernels arises for. Consequently, a uniform comparability of Artin patterns is guaranteed by the restriction for all.
Finally, in the case of the parent definition with, we have and for any. The largest of these kernels arises for. Consequently, a uniform comparability of Artin patterns is guaranteed by the condition for all.
P 5.9. Note that the first and third condition coincide since both, and, denote the commutator subgroup. So the family is restricted to the normal subgroups which contain, as announced in the paragraph preceding Lemma 3.1.
The second condition restricts the family to the maximal subgroups of G inclusively the group G
itself and the Frattini subgroup.
P 5.10. Since we shall mainly be concerned with the first and third parent definition for descendant trees, that is, either with respect to the lower central series or to the derived series, the comparability condition in Theorem 5.3 suggests the definition of a category whose objects are subject to more severe conditions than those in Formula (5.23),
but whose morphism are defined exactly as in Formula (5.24). The new viewpoint leads to a corresponding modification of Artin transfer patterns.
Definition 5.6. Let be an object of the category.
The Artin pattern, more precisely the restricted Artin pattern, of G is the pair
whose components, the TTT and the TKT of G, are defined as in the Formulas (5.25) and (5.26), but now with respect to the smaller system of subgroups of G.
P 5.11. The following Main Theorem shows that any non-metabelian group G with derived length and finite abelianization shares its Artin transfer pattern, in the restricted sense, with its metabelianization, that is the second derived quotient.
Theorem 5.4. (Main Theorem.) Let G be a (non-metabelian) group with finite abelianization, and denote by, , the terms of the derived series of G, that is and for, in particular, and, then
1) every subgroup which contains the commutator subgroup is a normal subgroup of finite index,
2) for each, there is a chain of normal subgroups
3) for each, the targets of the transfers and
are equal in the sense of the natural order,
4) for each, the kernels of the transfers and
are equal in the sense of the natural order,
Proof. We use the natural epimorphism,.
1) If U is an intermediate group, then is a normal subgroup of G, according to Lemma 3.1. The assumption implies that is a divisor of the integer. Therefore, the Artin transfer exists.
2) Firstly, implies. Since is characteristic in U, we also have. Similarly, is characteristic in and thus normal in. Finally, we obtain
3) The mapping, , is an epimorphism with kernel. Consequently, the isomorphism theorem in Remark 7.3 of the appendix yields the isomorphism .
4) Firstly, the restriction is an epimorphism which induces an isomorphism
, since and
, according to Theorem 5.1. Secondly, according
to Theorem 5.2, the condition implies that the index is finite, the Artin transfer exists, the composite mappings commute, and, since we even have, the transfer kernels satisfy the relation
. In the sense of the natural partial order on transfer kernels this
means equality, since and thus, similarly as in Proposition 5.1, the map establishes a set bijection between the systems of subgroups and
Remark 5.1. At this point it is adequate to emphasize how similar concepts in previous publications are related to the concept of Artin patterns. The restricted Artin pattern in Definition 5.6 was essentially introduced in ( , Dfn.1.1, p. 403), for a special case already earlier in ( , §1, p. 417). The name Artin pattern appears in ( , Dfn.3.1, p. 747) for the first time. The complete Artin pattern in Definition 5.3 is new in the present article, but we should point out that it includes the iterated IPADs (index-p abelianization data) in ( , Dfn.3.5, p. 289) and the iterated IPODs (index-p obstruction data) in ( , Dfn.4.5).
In a second remark, we emphasize the importance of the preceding Main Theorem for arithmetical applications.
Remark 5.2. In algebraic number theory, Theorem 5.4 has striking consequences for the determination of the length of the p-class tower, that is the maximal unramified pro-p extension, of an algebraic number field K with respect to a given prime number p. It shows the impossibility of deciding, exclusively with the aid of the restricted Artin pattern, which of several assigned candidates G with distinct derived lengths is the actual p-class tower group. (In contrast, can always be recognized with.)
This is the point where the complete Artin pattern enters the stage. Most recent investigations by means of iterated IPADs of 2nd order, whose components are contained in, enabled decisions between in   .
Another successful method is to employ cohomological results by I.R. Shafarevich on the relation rank for selecting among several candidates G for the p-class tower group, in dependence on the torsion-free unit rank of the base field K, for instance in   .
Important examples for the concepts in §5 are provided in the following subsections.
5.5. Abelianization of Type (p,p)
Let G be a p-group with abelianization of elementary abelian type. Then G has maximal subgroups of index. For each, let be the Artin transfer homomorphism from G to the abelianization of.
Definition 5.7. The family of normal subgroups is called the transfer kernel type
(TKT) of G with respect to.
Remark 5.3. For brevity, the TKT is identified with the multiplet, whose integer components
are given by
Here, we take into consideration that each transfer kernel must contain the commutator subgroup of G, since the transfer target is abelian. However, the minimal case cannot occur, according to Hilbert’s Theorem 94.
A renumeration of the maximal subgroups and of the transfers by means of a per-
mutation gives rise to a new TKT with respect to, identified with, where
It is adequate to view the TKTs as equivalent. Since we have
the relation between and is given by. Therefore, is another representative of the orbit of under the operation of the symmetric group on the set of all mappings from to, where the extension of the permutation is defined by, and we formally put,.
Definition 5.8. The orbit of any representative is an invariant of the p-group G and is called its transfer kernel type, briefly TKT.
Remark 5.4. This definition of goes back to the origins of the capitulation theory and was introduced by Scholz and Taussky for in 1934  . Several other authors used this original definition and investigated capitulation problems further. In historical order, Chang in 1977  , Chang and Foote in 1980  , Heider and Schmithals in 1982  , Brink in 1984  , Brink and Gold in 1987  , Nebelung in 1989  , and ourselves in 1991  and in 2012  .
In the brief form of the TKT, the natural order is expressed by for.
Let denote the counter of total transfer kernels, which is
an invariant of the group G. In 1980, Chang and Foote  proved that, for any odd prime p and for any integer, there exist metabelian p-groups G having abelianization of type such that. However, for, there do not exist non-abelian 2-groups G with, such that. Such groups must be metabelian of maximal class. Only the elementary abelian 2-group has.
In the following concrete examples for the counters, and also in the remainder of this article, we use identifiers of finite p-groups in the SmallGroups Library by Besche, Eick and O’Brien   .
Example 5.1. For, we have the following TKTs:
・ for the extra special group of exponent 9 with,
・ for the two groups with,
・ for the group with,
・ for the group with,
・ for the extra special group of exponent 3 with.
5.6. Abelianization of Type (p2, p)
Let G be a p-group with abelianization of non-elementary abelian type. Then G possesses maximal subgroups of index, and subgroups of index. See Figure 1.
P 5.12. Convention. Suppose that is the distinguished maximal subgroup which is the product of all subgroups of index, and is the distinguished subgroup of index
which is the intersection of all maximal subgroups, that is the Frattini subgroup of G.
P 5.13. First layer. For each, let be the Artin transfer homomorphism from G to the ab- elianization of.
Definition 5.9. The family is called the first layer transfer kernel type of G with respect to and, and is identified with, where
Remark 5.5. Here, we observe that each first layer transfer kernel is of exponent p with respect to and consequently cannot coincide with for any, since is cyclic of order, whereas is bicyclic of type.
P 5.14. Second layer. For each, let be the Artin transfer homomorphism from G to the abelianization of.
Definition 5.10. The family is called the second layer transfer kernel type of G with respect to and, and is identified with, where
P 5.15. Transfer kernel type.
Combining the information on the two layers, we obtain the (complete) transfer kernel type
of the p-group G with respect to and.
Remark 5.6. The distinguished subgroups and are unique invariants of G and should not be renumerated. However, independent renumerations of the remaining maximal subgroups and the transfers by means of a permutation, and of the remaining subgroups of index and the transfers by means of a permutation
, give rise to new TKTs with respect to and, identified with, where
and with respect to and, identified with,
It is adequate to view the TKTs and as equivalent. Since we have
the relations between and, resp. and, are given by, resp.. Therefore, is another representative of the orbit of under the operation
of the product of two symmetric groups on the set of all pairs of mappings from to, where the extensions and of a permutation are defined by and, and formally
, , , and.
Definition 5.11. The orbit of any representative is an invariant of the p- group G and is called its transfer kernel type, briefly TKT.
P 5.16. Connections between layers.
The Artin transfer from G to a subgroup of index
is the compositum of the induced transfer from to
(in the sense of Corollary 7.1 or Corollary 7.3 in the appendix) and the Artin transfer from G to, for any intermediate subgroup of index (). There occur two situations:
・ For the subgroups only the distinguished maximal subgroup is an intermediate subgroup.
・ For the Frattini subgroup all maximal subgroups are intermediate subgroups.
This causes restrictions for the transfer kernel type of the second layer,
since, and thus
・ , for all,
・ but even.
Furthermore, when with and, an element () which is of order with respect to, can belong to the transfer kernel only if its pth power is contained in, for all intermediate subgroups, and thus:
・ , for certain, enforces the first layer TKT singulet,
・ but, for some, even specifies the complete first layer TKT multiplet, that is, for all.
6. Stabilization and Polarization in Descendant Trees
P 6.1. Theorem 5.4 has proved that it suffices to get an overview of the restricted Artin patterns of metabelian groups G with, since groups G of derived length will certainly reveal exactly the same patterns as their metabelianizations.
In this section, we present the complete theory of stabilization and polarization of the restricted Artin patterns for an extensive exemplary case, namely for all metabelian 3-groups G with abelianization of type (3,3).
Since the bottom layer, resp. the top layer, of the restricted Artin pattern will be considered in Theorem 6.4 on the commutator subgroup, resp. Theorem 6.5 on the entire group G, we first focus on the intermediate layer of the maximal subgroups.
6.1. 3-Groups of Non-Maximal Class
P 6.2. We begin with groups G of non-maximal class. Denoting by m the index of nilpotency of G, we let with be the centralizers of two-step factor groups of the lower central series, that is, the biggest subgroups of G with the property. They form an as- cending chain of characteristic subgroups of G, , which contain the commutator subgroup. coincides with G if and only if. We characterize the smallest two-step centralizer different from the commutator group by an isomorphism invariant . According to Nebelung  , we can assume that G has order, class, and coclass, where. Let generators of be selected such that, , if, and. Suppose that a fixed ordering of the four maximal subgroups of G is defined by with, , , and. Let the main commutator of G be declared by and higher commutators recursively by, for. Starting with the powers, , let, for, and put,.
Theorem 6.1. (Non-maximal class.) Let G be a metabelian 3-group of nilpotency class and coclass with abelianization. With respect to the projection onto the parent, the restricted Artin pattern of G reveals
1) a bipolarization and partial stabilization, if G is an interface group with bicyclic last lower central equal to the bicyclic first upper central, more precisely
2) a unipolarization and partial stabilization, if G is a core group with cyclic last lower central and bicyclic first upper central, more precisely
3) a nilpolarization and total stabilization, if G is a core group with cyclic last lower central equal to the cyclic first upper central, more precisely
Proof. Theorems 5.1 and 5.2 tell us that for detecting whether stabilization occurs from parent to child G, we have to compare the projection kernel with the commutator subgroups of the four maximal normal subgroups,. According to ( , Cor.3.2, p. 480) these derived subgroups are given by
provided the generators of G are selected as indicated above. On the other hand, the projection kernel is given by
Combining this information with, we obtain the following results.
・ for if, independently of.
・ for if, which implies.
・ but if, , which also implies.
・ for if, , which implies .
Taken together, these results justify all claims. □
Example 6.1. Generally, the parent of an interface group G ( , Dfn.3.3, p. 430) with bicyclic last non-trivial lower central is a vertex of a different coclass graph with lower coclass. In the case of a bipolarization ( , Dfn.3.2, p. 430), which is now also characterized via the Artin pattern by Formula (6.1) for, we can express the membership in coclass graphs by the implication: If with, then. A typical example is the group of coclass 3 with parent of coclass 2 (again with identifiers in the SmallGroups database   ), where
In contrast, a core group G ( , Dfn.3.3, p. 430) with cyclic last non-trivial lower central and its parent are vertices of the same coclass graph. In dependence on the p-rank of its centre, the Artin pattern either shows a unipolarization as in Formula (6.2), if the centre is bicyclic, or a total stabilization as in Formula (6.3), if the centre is cyclic. Typical examples are the group with parent, both of coclass 2, where the Artin pattern shows a unipolarization
and the group with parent, both of coclass 2, where the Artin pattern shows a total stabilization
6.2. p-Groups of Maximal Class
P 6.3. Next we consider p-groups of maximal class, that is, of coclass, but now for an arbitrary prime number. According to Blackburn  and Miech  , we can assume that G is a metabelian p-group of order and nilpotency class, where. Then G is of coclass and the commutator factor group of G is of type. The lower central series of G is defined recursively by and for, in particular.
The centralizer of the two-step factor group
, that is,
is the biggest subgroup of G such that. It is characteristic, contains the commutator subgroup, and coincides with G, if and only if. Let the isomorphism invariant of G be defined by
where for, for, and for, according to Miech ( , p. 331).
Suppose that generators of are selected such that, if, and.
We define the main commutator and the higher commutators for.
The maximal subgroups of G contain the commutator subgroup of G as a normal subgroup of index p and thus are of the shape. We define a fixed ordering by and for.
Theorem 6.2. (Maximal class.) Let G be a metabelian p-group of nilpotency class and coclass, which automatically implies an abelianization of type. With respect to the projection onto the parent, the restricted Artin pattern of G reveals
1) a unipolarization and partial stabilization, if the first maximal subgroup of G is abelian, more precisely
2) a nilpolarization and total stabilization, if all four maximal subgroups of G are non-abelian, more precisely
In both cases, the commutator subgroups of the other maximal normal subgroups of G are given by
Proof. We proceed in the same way as in the proof of Theorem 6.1 and compare the projection kernel with the commutator subgroups of the maximal normal subgroups,. According to ( , Cor.3.1, p. 476) they are given by
if the generators of G are chosen as indicated previously. The cyclic projection kernel is given uniformly by
Using the relation, we obtain the following results.
・ for if.
・ if and only if, that is,.
The claims follow by applying Theorems 5.1 and 5.2. □
Example 6.2. For, typical examples are the group with parent, both of coclass 1, where the Artin pattern shows a unipolarization ( , Dfn.3.1, p. 413)
and the group with parent, both of coclass 1, where the Artin pattern shows a total stabilization
6.3. Extreme Interfaces of p-Groups
P 6.4. Finally, what can be said about the extreme cases (excluded in Theorems 6.1 and 6.2) of non-abelian p-groups having the smallest possible nilpotency class for coclass and for coclass? In these particular situations, the answers can be given for arbitrary prime numbers.
Theorem 6.3. Let G be a metabelian p-group with abelianization of type.
1) If G is of coclass and nilpotency class, then must be odd and the coclass must be exactly.
2) If G is of coclass and nilpotency class, then G is an extra special p-group of order and exponent p or.
In both cases, there occurs a total polarization and no stabilization at all, more explicitly
Proof. Suppose that G is a metabelian p-group with.
1) According to O. Taussky  , a 2-group G with abelianization of type must be of coclass. Consequently, implies.
Since the minimal nilpotency class c of a non-abelian group with coclass is given by, the case cannot occur for.
So we are considering metabelian p-groups G with, nilpotency class and coclass for odd, which form the stem of the isoclinism family in the sense of P. Hall. According to ( , Lem.3.1, p. 446), the commutator subgroups of the maximal subgroups are cyclic of degree p, for such a group. However, the kernel of the parent projection is the bicyclic group of type ( , §3.5, p. 445), which cannot be contained in any of the cyclic with.
2) According to ( , Cor.3.1, p. 476), the commutator subgroups of all maximal subgroups are trivial, for a metabelian p-group G of coclass and nilpotency class, which implies. Thus, the kernel of the parent projection is not contained in any.
In both cases, the final claim is a consequence of the Theorems 5.1 and 5.2. □
Example 6.3. For, a typical example for the interface between groups of coclass 2 and 1 is the group of coclass 2 with parent of coclass 1, where the Artin pattern shows a total polarization
For, a typical example for the interface between non-abelian and abelian groups is the extra special quaternion group with parent both of coclass 1, where the Artin pattern shows a total polarization
Summarizing, we can say that the last three Theorems 6.1, 6.2, and 6.3 underpin the fact that Artin transfer patterns provide a marvellous tool for classifying finite p-groups.
6.4. Bottom and Top Layer of the Artin Pattern
P 6.5. We conclude this section with supplementary general results concerning the bottom layer and top layer of the restricted Artin pattern.
Theorem 6.4. (Bottom layer.) The type of the commutator subgroup can never remain stable for a metabelian vertex of a descendant tree with respect to the lower central series, lower exponent-p central series, or derived series. The kernel of is equal to G (Principal Ideal Theorem).
Proof. All possible kernels , resp. , resp. , of the parent projections are non trivial, and can therefore never be contained in the trivial second derived subgroup G". According to Theorem 5.1, the type of the commutator subgroup G' cannot be stable. The Principal Ideal Theorem is due to Furtwängler  and is also proved in ( Thm.10.18, p. 313). □
Example 6.4. In Example 6.1, we point out that the group with cyclic centre and its parent , both of coclass 2, cannot be distinguished by their TTT
due to a total stabilization of the restricted Artin pattern as in Formula (6.3). However, the type of their commutator subgroup (the second layer of their TTT) admits a distinction, since
Theorem 6.5. (Top layer.) In a descendant tree with respect to the lower central series or derived series, the type of the abelianization of remains stable. The kernel of is equal to .
Proof. This follows from Theorem 5.1, since even the maximal possible kernel , resp. , of the parent projections is contained in the commutator subgroup of G.
We briefly emphasize the different behaviour of trees where parents are defined with the lower exponent-p central series.
Theorem 6.6. In a descendant tree with respect to the lower exponent-p central series, only the p-rank of the abelianization of the vertices remains stable.
Proof. Denote by the p-rank of the abelianization of G. According to Theorem 5.1, the maximal possible kernel of the parent projections is the Frattini subgroup which is contained in all maximal subgroups of G. According to Proposition 5.1, the map induces a bijection between the sets of maximal subgroups of the child G and the parent , whose cardinality is given by
. Consequently, we have . □
The author would like to express his heartfelt gratitude to Professor Mike F. Newman from the Australian National University in Canberra, Australian Capital Territory, for his continuing encouragement and interest in our endeavour to strengthen the bridge between group theory and class field theory which was initiated by the ideas of Emil Artin, and for his untiring willingness to share his extensive knowledge and expertise and to be a source of advice in difficult situations.
We also gratefully acknowledge that our research is supported by the Austrian Science Fund (FWF): P 26008- N25.
Appendix: Induced Homomorphism between Quotient Groups
Throughout this appendix, let be a homomorphism from a source group (domain) G to a target group (codomain) H.
A.1. Image, Pre-Image and Kernel
P 7.1. First, we recall some basic facts concerning the image and pre-image of normal subgroups and the kernel of the homomorphism .
Lemma 7.1. Suppose that and are subgroups, and are elements.
1) If is a normal subgroup of G, then its image is a normal subgroup of the (total) image .
2) If is a normal subgroup of the image , then the pre-image is a normal subgroup of G.
In particular, the kernel of is a normal subgroup of G.
3) If , then there exists an element such that .
4) If , then , i.e., the pre-image of the image satisfies
5) Conversely, the image of the pre-image is given by
The situation of Lemma 7.1 is visualized by Figure 4, where we briefly write and .
Remark 7.1. Note that, in the first statement of Lemma 7.1, we cannot conclude that is a normal subgroup of the target group H, and in the second statement of Lemma 7.1, we need not require that is a normal subgroup of the target group H.
Proof. 1) If , then for all ,
and thus for all , i.e., .
2) If , then , that is, . In particular, we have , i.e., , and
To prove the claim for the kernel, we put .
3) If , then , and thus . (See also  , Thm.2.2.1, p. 27).
4) If , then , and thus , by (3). This shows , and the opposite inclusion is obvious.
Figure 4. Kernel, image and pre-image under a homomorphism f.
Finally, since is normal, we have .
5) This is a consequence of the properties of the set mappings and associated with the homomorphism .
A.2. Criteria for the Existence of the Induced Homomorphism
P 7.2. Now we state the central theorem which provides the foundation for lots of useful applications. It is the most general version of a series of related theorems, which is presented in Bourbaki ( , Chap.1), Structures algëriques, Prop.5, p. A I.35]. Weaker versions will be given in the subsequent corollaries.
Theorem 7.1. (Main Theorem)
Suppose that is a normal subgroup of G and is a normal subgroup of H. Let and denote the canonical projections onto the quotients.
・ The following three conditions for the homomorphism are equivalent.
1) There exists an induced homomorphism such that , that is,
・ If the induced homomorphism of the quotients exists, then it is determined uniquely by , and its kernel, image and cokernel are given by
In particular, is a monomorphism if and only if .
Moreover, is an epimorphism if and only if .
In particular, is certainly an epimorphism if is onto.
We summarize the criteria for the existence of the induced homomorphism in a formula:
The situation of Theorem 7.1 is shown in the commutative diagram of Figure 5.
Remark 7.2. If the normal subgroup in the assumptions of Theorem 7.17 is taken as , then the induced homomorphism exists automatically and is a monomorphism.
Figure 5. Induced homomorphism of quotients.
Note that does not imply but only , if is not an epimorphism. Similarly, does not imply but only , if is not a monomorphism.
・ (1) Þ (2): If there exists a homomorphism such that for all , then, for any , we have , and thus , which means . It follows that .
(2) Þ (1): If , then the image of the coset under is independent of the re- presentative : If for , then and thus . Consequently, we have . Furthermore, is a homomorphism, since
(2) Þ (3): If , then .
(3) Þ (2): If , then .
・ The image of any under is determined uniquely by , since .
The kernel of is given by , and for we have
that is , which clearly contains , since .
The cokernel of is given by , if .
Finally, if is an epimorphism, then is also an epimorphism, which forces the terminal map to be an epimorphism. □
A.3. Factorization through a Quotient
P 7.3. Theorem 7.1 can be used to derive numerous special cases. Usually it suffices to consider the quotient group corresponding to a normal subgroup U of the source group G of the homomorphism and to view the target group H as the trivial quotient H/1. In this weaker form, the existence criterion for the induced homomorphism occurs in Lang’s book ( , p. 17).
Corollary 7.1. (Factorization through a quotient)
Suppose is a normal subgroup of G and denotes the natural epimorphism onto the quotient.
If , then there exists a unique homomorphism such that , that is, for all .
Moreover, the kernel of is given by .
Again we summarize the criterion in a formula:
In this situation the homomorphism is said to factor or factorize through the quotient via the canonical projection and the induced homomorphism .
The scenario of Corollary 7.1 is visualized by Figure 6.
Proof. The claim is a consequence of Theorem 7.1 in the special case that is the trivial group. The equivalent conditions for the existence of the induced homomorphism are resp. . □
Remark 7.3. Note that the well-known isomorphism theorem (sometimes also called homomorphism theorem) is a special case of Corollary 7.1. If we put and if we assume that is an epimorphism with , then the induced homomorphism is an isomorphism, since .
In this weakest form,
Figure 6. Homomorphism f factorized through a quotient.
actually without any additional assumptions being required, the existence theorem for the induced homomorphism appears in almost every standard text book on group theory or algebra, e.g., ( , Thm.2.3.2, p. 28) and ( , Thm.X.18, p. 339).
A.4. Application to Series of Characteristic Subgroups
P 7.4. The normal subgroup in the assumptions of Corollary 7.1 can be specialized to various characteristic subgroups of G for which the condition can be expressed differently, namely by invariants of series of characteristic subgroups.
Corollary 7.2. The homomorphism can be factorized through various quotients of G in the following way. Let n be a positive integer and p be a prime number.
1) factors through the nth derived quotient if and only if the derived length of is bounded by .
2) factors through the nth lower central quotient if and only if the nilpotency class of is bounded by .
3) factors through the nth lower exponent-p central quotient if and only if the p-class of is bounded by .
We summarize these criteria in terms of the length of series in a formula:
Proof. By induction, we show that, firstly,
Now, the claims follow from Corollary 7.1 by observing that iff, iff, and iff □
The following special case is particularly well known. Here we take the commutator subgroup of G as our charecteristic subgroup, which can either be viewed as the term of the lower central series of G or as the term of the derived series of G.
Corollary 7.3. A homomorphism passes through the derived quotient of its source group G if and only if its image is abelian.
Proof. Putting in the first statement or in the second statement of Corollary 7.2 we obtain the well-known special case that passes through the abelianization if and only if is abelian, which is equivalent to, and also to. □
The situation of Corollary 7.3 is visualized in Figure 7.
Using the first part of the proof of Corollary 7.2 we can recognize the behavior of several central series under homomorphisms.
Lemma 7.2. Let be a homomorphism of groups and suppose that is an integer and a prime number. Let be a subgroup with image.
1) If, then
2) If, then
3) If, then
Proof. 1) Let, then and
. Consequently, we have ifand if.
2) Let, then and. Thus, we have ifand if.
3) Let, then and. Therefore, we have ifand if.□
A.5. Application to Automorphisms
Corollary 7.4. (Induced automorphism)
Figure 7. Homomorphism passing through the derived quotient.
Let be an epimorphism of groups, , and assume that is an auto- morphism of G.
1) There exists an induced epimorphism such that, if and only if, resp..
2) The induced epimorphism is also an automorphism of H, , if and only if
In the second statement, is said to have the kernel invariance property (KIP) with respect to.
The situation of Corollary 7.4 is visualized in Figure 8.
Proof. Since is supposed to be an epimorphism, the well-known isomorphism theorem in Remark 7.3 yields a representation of the image as a quotient.
1) According to Theorem 7.1, the automorphism, simply viewed as a homomorphism, induces a homomorphism if and only if. Since is an epimorphism, is also an epimorphism with kernel.
Remark 7.4. If is a characteristic subgroup of G, then Corollary 7.4 makes sure that any automorphism induces an automorphism, where. The reason is that, by definition, a characteristic subgroup of G is invariant under any automorphism of G.
P 7.5. We conclude this section with a statement about GI-automorphisms (generator-inverting auto- morphisms) which have been introduced by Boston, Bush and Hajir ( , Dfn.2.1). The proof requires results of Theorem 7.1, Corollary 7.4, and Corollary 7.2.
Theorem 7.2. (Induced generator-inverting automorphism)
Let be an epimorphism of groups with, and assume that is an automorphism satisfying the KIP, and thus induces an automorphism.
If is generator-inverting, that is,
then is also generator-inverting, that is, for all.
Proof. According to Corollary 7.4,
induces an automorphism, since.
Two applications of the Remark 7.4 after Corollary 7.4, yield:
induces an automorphism, since is characteristic in G, and
induces an automorphism, since is characteristic in H.
Using Theorem 7.1 and the first part of the proof of Corollary 7.2, we obtain:
induces an epimorphism, since.
The actions of the various induced homomorphisms are given by
Figure 8. Induced automorphism.
Finally, combining all these formulas and expressing for a suitable, we see that implies the required relation for:
A.6. Functorial Properties
P 7.6. The mapping which maps a homomorphism of one category to an induced homomorphism of another category can be viewed as a functor F.
In the special case of induced homomorphisms between quotient groups, we define the domain of the functor F as the following category.
The objects of the category are pairs consisting of a group G and a normal subgroup,
For two objects, the set of morphisms consists of homomorphisms such that, briefly written as arrows,
The functor from this new category to the usual category of groups
maps a pair to the corresponding quotient group,and it maps a morphism to the induced homomorphism ,
Existence and uniqueness of have been proved in Theorem 7.1 under the assumption that, which is satisfied according to the definition of the arrow.
The functorial properties, which are visualized in Figure 9, can be expressed in the following form.
Figure 9. Functorial properties of induced homomorphisms.
Firstly, F maps the identity morphism having the trivial property to the identity homomorphism
and secondly, F maps the compositum of two morphisms
and, which obviously enjoys the required property
to the compositum
of the induced homomorphisms in the same order.
The last fact shows that F is a covariant functor.
*Respectfully dedicated to Professor M. F. Newman.