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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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The Generalization of the Stallings Theorem

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, Iran
Tel:
982144865179
E-mail:
[email protected]
 
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.

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Abstract

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.

Keywords

Schur-Baer variety; Pair of groups; image
G-marginal series

Introduction

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 image
, where M(G) is the Schur multiplier of G, image
is the second cohomology of G with coefficient in image
is the second internal homology of G [2]. Hopf [3] proved that image
. 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

image

where H3(−) 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 image
. 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 image
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 = {x1, x2,…} and image
and image
be two varieties of groups defined by the sets of laws image
and image
, 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:

image

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 image
and image
.

Let G be an arbitrary group in image
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 image
-free presentation of the group G. We call

image

the generalized Baer-invariant of the group G in image
with respect to the variety image
. Now if N is a normal subgroup of the group G for a suitable normal subgroup S of the free group F, we have image
Then we can define the generalized Baer-invariant of the pair of groups with respect to two varieties image
and image
as follows:

image

One may check that image
image
M(G, N) is always abelian and independent of the free presentation of G. In particular, if image
is the variety of all groups and N=G then the generalized Baer-invariant of the pair (G, N) will be

image

which is the usual Baer-invariant of G with respect to image
[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 image
. In the next section, we show that under some circumstances there are some isomorphisms among image
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 image
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, image
M(G) is finite with respect to a Schur-Baer variety image
. In the following lemma we prove similar result for the image
image
M(G, N) and image
image
M(G) with using another technique.

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

image

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 image
, then

image
image
image

Also, image
Thus the result holds.

Stallings’ Theorem

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 , image
Then the following sequences are exact:

image
image

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

image

Proof. Considering the definition mentioned above we can conclude:

image

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

(iii) Using the assumption, we have image
. Therefore, one can easily check that the following sequence is exact:

image

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

image

where image
for all n ≥ 1. We call such a series the lower image
marginal series of N in G. One may also define the upper image
G- marginal series as in studies of Moghaddam et al. [11].

We say that the normal subgroup N of G is image
G-nilpotent if it has a finite lower image
marginal series. The shortest length of such series is called the class of image
nilpotency of N in G. If N = G, then this is called lower image
-marginal series of G. The group G is said to be image
-nilpotent iff Vn(G) = 1, for some positive integer n [12].

Now, we want to show that under some circumstances there are some isomorphisms among image
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 f0 :G / N → H / K and image
and thatimage
is an epimorphism. Then f induces isomorphismsimage
andimage
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 image
We proceed by induction. For n = 0 the assertion is trivial. For n = 1, consider the following diagram:

image

By the hypothesis image
are isomorphism, hence f1 is an isomorphism. Assume that n ≥ 2. By consedering Lemma 2.1(ii), we can conclude the following communicative diagram:

image

Note that the naturality of the map f induces homomorphisms αi, i = 1,2,…,5 such that image
is commutative. By hypothesis α1 is an epimorphism and α4, α5 are isomorphisms. Also, by considering the induction hypothesis and definition of the Baer-invariant of the pair of groups, α2 is an isomorphism. Hence by five lemma of Rotman’s studies [14] α3 is an isomorphism. Now consider the following diagram and in the same way, fn is an isomorphism.

Now we obtain the following corollary.

image

By the above discussion α3 is an isomorphism and by induction of hypothesis image
is an isomorphism, therefore, image
is an isomorphism. Finally, by the following diagram:

image

And the same way, fn 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 image
G-nilpotent and image
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 image

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

Theorem 2.4. Let image
be a variety of groups and f : G → H be an epimorphism. Let N be a image
G-nilpotent normal subgroup of G and K be a normal subgroup of H such that f (N) = K. If ker f ⊆[NV*G] and image
image
M(H, K) is trivial, then f and f | are isomorphisms.

Proof. Put M = ker f, then image
and image
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 thatimage
We denote by image
the intersection of all subgroups of the form image
. Clearly, image
is a characteristic subgroup of G which is contained in image
In particular, if image
is the variety of all groups and image
is a variety of abelian groups then this subgroup is denoted by image
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 image
Then

image

Proof. By theorem 3.2 of Neumann [5], natural homomorphism image
will be a monomorphism. Now the following commutative diagram

image

implies that the natural homomorphism image
also a monomorphism. Thus Lemma 1.2 (i) implies thatimage
is trivial. Now we have image
which completes the result.

Acknowledgement

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.

References

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