Medical, Pharma, Engineering, Science, Technology and Business

**Onsory A ^{*} and Araskhan M^{*}**

Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd, Iran

- Corresponding Author:
- Onsory A

Department of Mathematics

Yazd Branch, Islamic Azad

University, Yazd, Iran982144865179

Tel:[email protected]

E-mail:

- Araskhan M

Department of Mathematics, Yazd

Branch Islamic Azad University, Yazd, Iran

**Tel:**982144865179

**E-mail:**[email protected] iauyazd.ac.ir

**Received date:** November 28, 2015; **Accepted date:** December 21, 2015; **Published date:** December 23, 2015

**Citation:** Onsory A, Araskhan M (2015) The Generalization of the Stalling’s Theorem. J Generalized Lie Theory Appl S2:003. doi:10.4172/generalized-theory-applications.S2-003

**Copyright:** © 2015 Onsory A, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

In this paper, we present a relative version of the concept of lower marginal series and give some isomorphisms among ï��G-marginal factor groups. Also, we conclude a generalized version of the Stalling’s theorem. Finally, we present a sufficient condition under which the order of the generalized Baer-invariant of a pair of finite groups divides the order of the generalized Baer-invariant of its factor groups.

Schur-Baer variety; Pair of groups;

G-marginal series

There exists a long history of interaction between Schur multipliers and other mathematical concepts. This basic notion started by Schur [1], when he introduced multipliers in order to study projective representations of groups. It was known later that the Schur multiplier had a relation with **homology **and **cohomology **of groups. In fact, if G is a finite group, then

, where M(G) is the Schur multiplier of G,

is the second cohomology of G with coefficient in

is the second internal homology of G [2]. Hopf [3] proved that

. He also proved that the Schur multiplier of G is independent of the free presentation of G. Let (G, N) be a pair of groups, where N is a normal subgroup in Ellis [4] defined the Schur multiplier of the pair (G, N) to be the **abelian group **M(G, N) appears in the following natural exact sequence

where H_{3}(−) denote the third homology of a group with integer coefficients. He also proved that if the normal subgroup N possess a complement in G, then for each free presentation 1→ R→ F →G→1 of G, M(G, N) is isomorphic with the factor group (R∩[S, F]) / [R, F] , where S is a normal subgroup of F such that

. In particular, if N = G then the Schur multiplier of (G, N) will be M(G) = (R∩[F, F]) / [R, F].

*We assume that the reader is familiar with the notions of the verbal subgroup V(G), and the marginal subgroup*

V *(G), associated with a variety of groups

and a group G [5] for more information on varieties of groups). Let F_{∞} be the free group freely generated by the countable set X = {x_{1}, x_{2},…} and

and

be two varieties of groups defined by the sets of laws

and

, respectively. Let N be a normal subgroup of a group G, then we define [NV *G] to be the subgroup of G generated by the elements of the following set:

It is easily checked that [NV *G] is the least normal subgroup T of G such that N/T is contained in V *(G/T) [6].

The first to create the generalization of the Schur multiplier to any variety of groups was Baer [7]. It is well known fact that the recent concept is useful in classifying groups into isologism classes. Leedham- Green and McKay [8] introduced the following generalized version of the Baer-invariant of a group with respect to two varieties

and

.

Let G be an **arbitrary group **in

with a free presentation 1→ R→ F →G →1, in which F is a free group. Clearly, 1=W(G) =W(F )R / R and hence W(F ) ⊆ R , therefore,

1→ R /W(F )→ F /W(F )→G →1

is a

-free presentation of the group G. We call

the *generalized Baer-invariant* of the group G in

with respect to the variety

. Now if N is a normal subgroup of the group G for a suitable normal subgroup S of the free group F, we have

Then we can define the generalized Baer-invariant of the pair of groups with respect to two varieties

and

as follows:

One may check that

M(G, N) is always abelian and independent of the free presentation of G. In particular, if

is the variety of all groups and N=G then the generalized Baer-invariant of the pair (G, N) will be

which is the usual Baer-invariant of G with respect to

[8].

It is interesting to know the connection between the Baer-invariant of a pair of finite groups (G, N) and its factor groups with respect to the Schur-Baer variety

. In the next section, we show that under some circumstances there are some isomorphisms among

_{G}-marginal factor groups (Theorem 2.2). Also, a sufficient condition will be given such that the order of the generalized Baer-invariant of a pair of finite groups divides the order of the generalized Baer-invariant of the pair of its factor groups (Theorem 2.5).

Variety

is called a *Schur-Baer* variety if for any group G in which the marginal factor group G / V*(G) is finite, then the verbal subgroup V(G) is also finite. Schur [9] proved that the variety of abelian groups is a Schur-Baer variety and Baer [10] showed that a variety defined by outer commutator words carries this property. In 2002, Moghaddam et al. [11] proved that for a finite group G,

M(G) is finite with respect to a Schur-Baer variety

. In the following lemma we prove similar result for the

M(G, N) and

M(G) with using another technique.

**Lemma 1.1.** *Let
be a Schur-Baer variety and G be a finite group in
with a normal subgroup N. Then there exists a group H with a normal subgroup K such that*

Proof. Let G = F / R be a free presentation for the group G and S be a normal subgroup of the free group F such that

, then

Also,

Thus the result holds.

In the following lemma we present some exact sequences for the generalized Baer-invariant of a pair of groups and its factor groups.

**Lemma 2.1. ***Let G be a group with a free presentation 1→ R→ F →G→1 and S, T be normal subgroups of the free group F such that T ⊆ S ,
Then the following sequences are exact:*

(iii) Moreover, if K is contained in V*(G), then the following sequence is exact:

*Proof.* Considering the definition mentioned above we can conclude:

Now one can easily check that the sequences (i) and (ii) are exact.

(iii) Using the assumption, we have

. Therefore, one can easily check that the following sequence is exact:

Let N be a normal subgroup of a group G. Then we define a series of normal subgroups of N as follows:

where

for all n ≥ 1. We call such a series the lower

marginal series of N in G. One may also define the upper

_{G}- marginal series as in studies of Moghaddam et al. [11].

We say that the normal subgroup N of G is

G-*nilpotent* if it has a finite l*ower*

*marginal series*. The shortest length of such series is called the class of

nilpotency of N in G. If N = G, then this is called lower

-*marginal series* of G. The group G is said to be

-nilpotent iff V_{n}(G) = 1, for some positive integer n [12].

Now, we want to show that under some circumstances there are some isomorphisms among

marginal factor groups. By using Lemma 2.1, we have the following Theorem, which generalizes 7.9.1 of literature of Hilton and Stammbach [13].

**Theorem 2.2. ***Let f : G → H be a group homomorphism and N be a normal subgroup of G and K be a normal subgroup of H such that f (N) ⊆ K . Suppose f induces isomorphisms f _{0} :G / N → H / K and
and that
is an epimorphism. Then f induces isomorphisms
and
for all n ≥ 0.*

Proof. At first, we want to mention a point that for making it easier to draw the following diagrams, we would like to introduce

We proceed by induction. For n = 0 the assertion is trivial. For n = 1, consider the following diagram:

By the hypothesis

are **isomorphism**, hence *f _{1}* is an isomorphism. Assume that n ≥ 2. By consedering Lemma 2.1(ii), we can conclude the following communicative diagram:

Note that the naturality of the map f induces homomorphisms α_{i}, i = 1,2,…,5 such that

is commutative. By hypothesis α1 is an epimorphism and α_{4}, α_{5} are isomorphisms. Also, by considering the induction hypothesis and definition of the Baer-invariant of the pair of groups, α_{2} is an isomorphism. Hence by five lemma of Rotman’s studies [14] α_{3} is an isomorphism. Now consider the following diagram and in the same way,* f _{n}* is an isomorphism.

Now we obtain the following corollary.

By the above discussion α_{3} is an isomorphism and by induction of hypothesis

is an isomorphism, therefore,

is an isomorphism. Finally, by the following diagram:

And the same way, *f _{n}* ia an isomorphism.

Now we obtain the following collary.

**Corollary 2.3. ***Let ( f , f |) : (G, N)→(H,K) are group homomorphisms satisfy the hypotheses of Theorem 2.2. Suppose further that N and K are
_{G}-nilpotent and
_{H}-nilpotent, respectively. Then f and f | are isomorphisms.*

*Proof.* The assertion follows from Theorem 2.2 and the remark that there exists n ≥ 0 such that

Now, we have the following theorem, which is a generalization of **Stalling’s theorem **[15].

**Theorem 2.4.** *Let
be a variety of groups and f : G → H be an epimorphism. Let N be a
*

Proof. Put M = ker f, then

and

for all n ≥ 0. Now the result follows from Corollary 2.3.

Finally, a sufficient condition will be given such that the order of the generalized Baer-invariant of a pair of finite groups divides the order of the generalized Baer-invariant of the pair of its factor groups with respect to two varieties of groups. Let Finally, a sufficient condition will be given such that the order of the generalized Baer-invariant of a pair of finite groups divides the order of the generalized Baer-invariant of the pair of its factor groups with respect to two varieties of groups. Let ψ : E →G be an epimorphism such that

We denote by

the intersection of all subgroups of the form

. Clearly,

is a characteristic subgroup of G which is contained in

In particular, if *
* is the variety of all groups and

as in literature of Karpilovsky [2].

Now using the above concept we have the following Theorem.

**Theorem 2.5. ***Let K be a normal subgroup of G contained in
Then*

*Proof.* By theorem 3.2 of Neumann [5], natural homomorphism

will be a monomorphism. Now the following commutative diagram

implies that the natural homomorphism

also a monomorphism. Thus Lemma 1.2 (i) implies that

is trivial. Now we have

which completes the result.

The authors wish to thank Yazd Branch, Islamic Azad University for its support of research project under the title the Generalization of the Stalling’s theorem.

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