1. Introduction
Embedding is one of the most important properties of wreath product; this property was further investigated with regards to imprimitivity of groups, normal subgroups and Quotient Group. Many people have worked on wreath products over the years and their work is as shown below:
Suzuki [2] in 1982 proved the Kaloujnine-Krasner Theorem that states that if F is a group extension of N by G, then F can be embedded inside the standard wreath product . Audu [1] in 1991 proved that a permutation group that is transitive and imprimitive that is acting on a finite set can be embedded inside the wreath product , where is the setwise stabilizer of G and are the constituents of X and G respectively while Δ is an element in the set of imprimitivity. Dixon & Mortimer [3] in 1996 expounded that any transitive and imprimitive group G can be embedded inside a wreath product in such a way that the kernel consists of the set of elements of the group which are mapped into the base group. Conway et al. [4] in 1998 gave an example of a group of degree 8 that is generated by , and with four block system namely: , , , . They showed that the group which is imprimitive can be embedded inside the wreath product .
Bamberg [5] in 2005 states that if G is any transitive imprimitive permutation group on a set Ω and Λ a G-invariant partition of Ω, if also Δ is an element of Λ and C the permutation group induced by the action of on Δ. If D is the group of permutations induced by G on Λ, then Ω may be identified with in such a way that G can be embedded into the wreath product in imprimitive action. Bamberg further states that if G is transitive but imprimitive group on a finite set Ω, then G can be embedded into the wreath product acting in imprimitive action, where Δ is a block for G, is the group induced by the action of setwise stabilizer on Δ, and n is the size of the orbit of Δ under G. If is also imprimitive, then embeds into a wreath product. As Ω is finite, the process can continue until an embedding of G into iterated wreath product of primitive groups was found. Chan [6] in 2006 proved that every faithful group action that is transitive and imprimitive is embeddable in a wreath product. Cameron [7] in 2013 showed that if H is a permutation group induced on a part by its setwise stabilizer and if K is the permutation group induced on the set of parts by the group G, then G is embedded in the wreath product .
Tamuli [8] in (1972) gave a new prove of the Universal Embedding Theorem and further proved that if N is a subgroup of a group H and all are subgroups of another group G, then G can be embedded inside the wreath product . Tamuli further proved that if the subgroup H has a transversal T which centralizes H in G, then the embedding is an extension of the diagonal embedding . Tamuli also proved that if Q is an amalgam of two subgroups A and B in which their intersection N is a normal subgroup of B, and if T is the transversal of N in B, then the amalgam Q can be embedded inside the wreath product . Dixon & Mortimer [3] (1996) stated and gave a new proof of the Universal Embedding Theorem that states: If G is an arbitrary group with a normal subgroup N, and the factor group of G by N, then is an embedding such that maps N onto , where B is the base group of . Mikaelian [9] in 2002 showed that every extension of a group G where the group product is the product variety that consists of all extensions of groups, if N is a normal subgroup and , then every extension of G can be isomorphically embedded into the wreath product . Hulpke [10] in 2004 proved that a transitive group G can be embedded inside the wreath product if M is a normal subgroup of G and A any subgroup of G.
Given isomorphism between two groups, knowing how the first group is isomorphic to a subgroup of the other groups helps us to know the structures being preserved. Since a wreath product is a group with many subgroups, it is easily seen that to be isomorphic to a group.
In this paper, we were able to give new proof of the theorem by Audu (1991) (see [1] ); we obtained the proof of the following: a group can be embedded into the wreath product of a factor group by a normal subgroup; the wreath product of two factor groups can be embedded into a group; when the abstract group in the Universal Embedding Theorem is a p-group, cyclic and simple, the embedding is an isomorphism.
2. Basic Definitions
An action of a group G on a non-empty set Ω is a map denoted by for all , such that
1) for all and all (1)
2) for all (2)
We then say that G acts onΩ.
If G and H are groups, then is a group called the Direct Product of G and H where and multiplication is defined by
(3)
If is the identity for G, and is the identity for H, then is the identity for
and (4)
(see details in [11] )
If Γ and Δ are nonempty sets, then we call to denote the set of all functions from Δ to Γ. In the case that C is a group, we turn into a group by defining product “pointwise”
(5)
for all and where the product in the right is in C.
Let C and D be groups and suppose D acts on the nonempty set Δ. Then the wreath product of C by D is defined with respect to this action is defined to be the semidirect product where D acts on the group via
(6)
for all , and and multiplication for all is given by
(7)
Clearly, (8)
(see details in [12] )
A homomorphism that is one-to-one (injective) is called an embedding: the group G “embeds” into H as a subgroup. If is not one-to-one, then it is a quotient. Note that if is an embedding, then and from the First Isomorphism Theorem, . Now as is a homomorphism, and so we conclude that in an embedding, G is isomorphic to a subgroup of H. In symbol .
3. Results
We now give an alternative proof to a theorem of Audu (1991) and also outline some propositions with their proofs. We proved embedding by showing that they are homomorphic and injective. We gave three conditions on the Universal Embedding Theorem (Dixon & Mortimer, 1996) when the group is a p-group and when the group is simple.
Theorem 1 (see [1] ): Let G be any transitive and imprimitivity group acting on a set Ω; let be a system of imprimitivity of G and Δ be an element of Λ. If then G can be embedded inside the wreath product .
Proof: Let be a homomorphism of G onto with kernel . Let be defined by
(9)
for each . If , then
(by (9))
(by (9))
Therefore, since it implies that . Therefore, . Thus lies in the kernel K. That is, . By the definition of wreath product, we can define a function for each , such that by
(10)
for all .
We claim that defines an embedding of G into with the function (10).
We seek to show that is a homomorphism and is injective, hence an embedding.
Take , then
(by (7))
Now since is a homomorphism,
(11)
By (10), can be expressed for all as follows:
(by (6) and (11))
(by (10))
(by (10))
(by (10))
Therefore, we have that
(12)
Hence (by (11) and (12)). Hence is a homomorphism.
Next, we show that is injective. Now since implies and , and so . Thus .
Example 1:
Let is a transitive group. Then is block of G. . And , . Thus which is a group of order 1024.
Proposition 2: Let G be an arbitrary subgroup with a normal subgroup N and be the natural homomorphism. Suppose that is a homomorphism then there is an embedding making the diagram
Figure 1. Commutativity diagram. Commute. .
Proof: If the diagram is to commute (see Figure 1), then we have for an arbitrary ( ) such that
(13)
and that is the only way it can be defined. First, we notice that (13) reposes only on the coset xN and not on the representativex. For if, , then , and .
Hence and so
hence
and so (13) defines a map.
Next, if we have a different element of G/N, say yN, then
(by (13))
(by (13))
Thus is a homomorphism. From (13), . Thus an embedding. Therefore .
Example 2:
Let . is a normal subgroup of G. The factor group . Then the wreath product
which is a group of order 72.
Proposition 3: Let G be an arbitrary group with a normal subgroup N, and put . Then there is an embedding such that maps K onto where B is the base group of . (Thus contains an isomorphic copy of every extension G of K by N.)
Proof: Let be the natural homomorphism of G onto K. Let be a set of right coset representatives of N in G such that
(14)
for each . If ,
(by (14))
(by (14))
Therefore, since it implies that . Therefore, . Thus lies in . That is, . By the definition of the wreath product, we can define a function for each , such that by
(15)
for all .
We claim that defines an embedding of G into with the function (15).
We seek to show that is an embedding.
Take , then
(by (7))
Now since is a homomorphism,
(16)
By (15), can be expressed for all as follows:
(by (15) and (16))
(by (15))
(by (15))
(by (15))
Therefore, we have that
(17)
Hence (by (16) and (17)). Hence is a homomorphism.
Next, we show that is injective. Now since implies and , and so .
Finally, lies in B when , and this happens exactly when . Thus .
Example 3:
Let and is a normal subgroup of G. Then . Then the wreath product
which is a group of order 24.
Proposition 4: Let G be any arbitrary group with a normal subgroup N and put . Then there is an embedding such that maps onto K, where B is the base group of and . (Thus contains an isomorphic copy of every extension G of K by K.)
Proof: Let be the natural homomorphism of G onto K with kernel N. Let be a set of right coset representatives of N in G such that
(18)
for each . If , then
(by (18))
(by (18))
Therefore, since it follows that . Therefore, . Thus lies in the kernel N, that is, . By the definition of wreath product, we can define a function for each , such that by
(19)
for all .
Now defines an embedding of into G with the function (18).
We seek to show that is a homomorphism and injective, hence an embedding.
Take , then
Now since is a homomorphism,
(20)
By (17), can be expressed for all as follows:
(by (19))
(by (19))
(by (19))
(by (6) and (20))
(by (6) and (20))
Therefore, we have
(21)
Hence (by (20) and (21)), showing that is a homomorphism.
Next, we show that is injective. Now since implies and so and , thus from Lagrange theorem, , i.e. as and .
Finally, lies in B when , and this happens exactly when .
Thus .
Example 4:
Let and . Then . Now . Then the wreath product which is a group of order 8.
Remark 5: If is an embedding, then where is the index of G by H. If the index , then is as isomorphism.
Theorem 6 (Dixon & Mortimer, 1996): (Universal Embedding Theorem)
Let G be an arbitrary group with a normal subgroup N, and put . Then there is an embedding such that maps N onto where B is the base group of . (Thus contains an isomorphic copy of every extension G of N by K.)
Condition 1: If G is of order p, i.e. , then or 1, by Lagrange’s theorem, then either or .
If , then , then and so . Thus , and being a homomorphism implies that and so .
If , then and so . Thus , and being a homomorphism implies that and so .
Thus is an isomorphism if or rather is an isomorphism if G is cyclic.
Condition 2: If the order of G is , then G is known to be an Abelian group and by Lagrange’s theorem ,p or 1.
If , then and so . Thus . And being a homomorphism implies that and so and . Therefore, is an isomorphism if .
If , then . Thus being a homomorphism implies that and so and the index.
If , then and so . Thus being a homomorphism implies that and so . Thus . Therefore, is an isomorphism if .
Condition 3: If G is a simple group, then either or .
If , then , then , and being a homomorphism implies that and so .
If , then , then , and being a homomorphism implies that and so .
Thus is an isomorphism if G is simple.
Example 5:
Let . Then the normal subgroup or . If , then . Thus . If , then . Thus . Thus an isomorphism as G is Cyclic, simple and its order is prime.
4. Discussion
Embedding is an important property of wreath product as it helps in preserving structures between groups. Under some conditions we have seen that the Universal embedding Theorem is an isomorphism.
5. Conclusion
In this paper, we were able to give a new proof of the theorem by Audu (1991), which proved that a group can be embedded into the wreath product of a factor group by a normal subgroup and also proved that a factor group can be embedded inside a wreath product and the wreath product of a factor group by a factor group can be embedded into a group. It was shown that when the abstract group in the universal embedding theorem is a p-group, cyclic and simple, the embedding is an isomorphism.
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