We dealt in  and  with Lie groups of bi-Möbius transformations defined on . The concept can be extended linearly to in the following way.
If then , , . Let us study the function defined by
where , , , , , .
Theorem 1. Let us denote and , . The function satisfies the following relations:
1) , for every
2) and , for every
3) , for every
4) , hence , for every
5) for every
6) and for every . Moreover, only if or and only if or .
It results that the composition law defines a structure of Abelian group on with the unit element and the inverse element of .
Proof. The proof requires only elementary computation. For (5) it is enough to show that for arbitrary and this results again after an elementary (although a little more tedious) computation. The relation (6) shows that by removing the elements and we obtain a subgroup of this group. The functions
are Möbius transformations, since as long as and , which has been postulated. Moreover, due to the fact that are injective and taking into account Theorem 1 (6), if and only if and if and only if . These Möbius transformations induce transformations of defined by
Theorem 2. The set of transformations endowed with the composition law
is an Abelian group having the identity element and such that the inverse element of is .
This group is isomorphic with , the isomorphism being given by the mapping . It makes a Lie group with analytic structure as n-dimensional complex differentiable manifold.
Proof. Indeed, if , then by Theorem 1 (6) , hence .
The commutativity results from: .
The identity element is , since .
The composition law is associative since: .
Finally, the inverse element of is since for every .
It is obvious that the mapping is bijective. Since are analytic functions in , the function is analytic in . Obviously, as complex n-dimensional manifold has an analytic structure and then the isomorphism makes from a Lie group with analytic structure as complex n-dimensional manifold.
We used    for the basic knowledge about Lie groups and their actions.
The actions by left and right translations of on itself are defined as: , respectively .
Theorem 1 implies that acts freely and transitively on itself by left and right translations.
2. Discrete Subgroups of G
Let be an arbitrary element and for every let us denote
where and .
Then, for every we have and using the formula (4), for every , in particular which is the identity element in . It results that the group generated by is a subgroup of . By Theorem 1 (2) we have that only if , hence for every we have and since for every , implies then for and , we have , therefore the elements of are all distinct.
Theorem 3. For every , the group generated by is a discrete subgroup of .
Proof. The case of is trivial. Suppose that for a given we would have then , which means that , contrary to the assumption.
Corollary 1. For every the subgroup acts freely and properly discontinuously on by left and by right translations.
3. Antianalytic Involutions of
Let be a non-empty subsequence of and let
Then the mapping defined by
is a fixed point free involution of in the sense that some of the mappings are fixed point free involutions of , while the others are the identity mapping. Then, it is true for itself that for every we have . Moreover, there is no for which , since this would imply for every j and if this would mean , which is absurd. Since are antianalytic self mappings of , we will say that is antianalytic.
Let us notice that the functions of the form we just listed are not the only antianalytic involutions of . If for , and we take the Möbius transformation which maps the unit disc onto itself, the unit circle onto itself and the exterior of the unit disc onto itself, we can prove:
Theorem 4. The function , where , is a fixed point free antianalytic involution of .
Proof. We have that
which shows that is antianalytic, since its complex conjugate is analytic. The equality implies , which is impossible since h is fixed point free, hence is fixed point free. Finally,
showing that is an involution.
We keep the notation for any antianalytic involution of constructed with the functions of this type by the method of the previous paragraph.
A given antianalytic involution and the identity mapping of form a group of transformations of .
Theorem 5. The quotient space can be endowed with a differentiable manifold structure, so that it becomes a non orientable differentiable manifold.
Proof. Indeed, an analytic atlas can be created on such that the local chart for any point is the identity on . Next, if for a we have when local charts can be used, where with for and otherwise and when and otherwise. Obviously, with such an atlas is a differentiable manifold for which every change of chart is a complex analytic function. The projection function , by which we have for every , induces a differentiable manifold structure on . Indeed, to every chart on corresponds a chart on , where and if then , hence . This structure is no more analytic since every change of charts is antianalytic. However, it is harmonic and therefore of class .
Theorem 6. For any fixed point free antianalytic involution of there is a partition of into two sets and such that if and only if . With the induced topology of the topological spaces and , as well as are homeomorphic under the projection .
Proof. We give a constructive proof. Let be arbitrary. Since is a fixed point free involution of we have that . Then there are disjoint open neighborhoods of and of such that if and only if . Let be arbitrary. We infer that . Indeed, supposing would imply that , contrary to the hypothesis. Analogously we find a contradiction supposing . Then there are open disjoint neighborhoods of and of such that if and only if . Moreover, we can take such that . In this way we can build two sequences of open sets and such that and are disjoint and if and only if . To make sure that the process ends in a countable number of steps, we can decide to take all the points such that their coordinates in are rational. Moreover, and are then maximal in the sense that does not contain any open set. The set with the trace atlas of is a complex differentiable manifold of dimension less than n. We repeat the process for and we find that there are two relatively open maximal sets and such that if and only if . Moreover, is a complex differentiable manifold of dimension less than that of and the process continues steps until we obtain dimension 0. Then is a countable set such that if and only if . Then a partition of into and such that if and only if is straightforward. Let us denote and . Then and are such that , and if and only if .
It is obvious that and are one-to-one and onto functions and if the topology of is chosen such that the projection is continuous, then the three topological spaces are homeomorphic.
We needed this construction for the following reason. The notation is ambiguous in the sense that on the right hand side we have an ordered couple of points, while in reality for the the order does not count and there is a situation which will appear later where this fact is essential. Now we can decide that once and have been built, they will remain permanently the same and every time we meet a couple we have chosen . It is as if we ignore occasionally the existence of and instead of we work only with .
We can define an operation on by using the composition law in from the section 1. For every couple and from we write , where .
Theorem 7. The multiplication is an internal composition law in with the unit element and , the inverse element of . The multiplication is commutative but not associative and therefore this law does not define a structure of Lie group on .
Proof. It is obvious that for every , is well defined and represents an element of . Moreover, , and . The non associativity of the law comes from the fact that although it may happen that and then the expression has no meaning.
We were expecting that one of the group axioms of the multiplication in not to be satisfied, since otherwise this manifold would be a Lie group and it is known (see , page 140) that every Lie group is an orientable manifold. However, as proven in the next theorem, actions of Lie groups on such a manifold exist.
Theorem 8. The mapping defined by is a left action of the Lie group on the non orientable manifold .
Proof. The mapping is obviously of class . Moreover, for every . Finally,
, hence indeed, is a left action of the Lie group on the non orientable manifold .
Corollary 2. The mapping defined by is a right action of on , since , as it can be easily checked.
By the general theory of Lie groups, these actions of on define homomorphisms t from the Lie group to the group of diffeomorphisms of such that is the mapping , respectively . Reciprocally, every homomorphism defines left and right actions of on by , respectively .
Non orientable n-dimensionl complex manifolds can be obtained by factorization with a two elements group generated by an antianalytic involution of . Such involutions can be obtained, for example, composing in some coordinate planes Möbius transformations of the form , with the mappings . An internal composition law can be defined on such a manifold with the help of some bi-Möbius transformations and actions of Lie groups on the respective manifold can be put into evidence. We realized this task by devising an appropriate partition of that manifold.
 Barza, I. and Ghisa, D. (1997) Lie Groups Actions on the Möbius Strip, Topics in Complex Analysis. In: Dimiev, S. and Sekigawa, K., Eds., Differential Geometry and Physics, World Sci., Singapore, 62-72.
 Barza, I. and Ghisa, D. (2020) Lie Groups Actions on Non Orientable Klein Surfaces. In: Dobrev V., Ed., Lie Theory and Its Applications in Physics, Springer, Singapore, Vol. 335, 421-428.