On Baer Invariants of Groups with Topological Approach (original) (raw)

Generalized Baer-Invariant of a Pair of Groups and the Direct Limit

journal of sciences islamic republic of iran, 2011

In this paper we introduce the concept of generalized Baer-invariant of a pair of groups with respect to two varieties ν and ω of groups. We give some inequalities for the generalized Baer-invariant of a pair of finite groups, when ν is considered to be the Schur-Baer variety. Further, 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 a pair of their factor groups. Moreover, we prove that the generalized Baer-invariant of a pair of groups commutes with direct limit.

On Baer Invariants of Pairs of Groups

2011

In this paper, we use the theory of simplicial groups to develop the Schur multiplier of a pair of groups (G,N)(G,N)(G,N) to the Baer invariant of it, mathcalVM(G,N)\mathcal{V}M(G,N)mathcalVM(G,N), with respect to an arbitrary variety mathcalV\mathcal{V}mathcalV. Moreover, we present among other things some behaviors of Baer invariants of a pair of groups with respect to the free product and the direct

Some Notes on the Baer-Invariant of a Nilpotent Product of Groups

Journal of Algebra, 2001

presented a formula for the Schur multiplier of a regular product of groups. In this paper, first, it is shown that the Baer-invariant of a nilpotent product of groups with respect to the variety of nilpotent groups has a homomorphic image and in finite case a subgroup of Haebich's type. Second, a formula will be presented for the Baer-invariant of a nilpotent product of cyclic groups with respect to the variety of nilpotent groups.

The Baer-invariant of a semidirect product

In 1972 K.I.Tahara [7,2 Theorem 2.2.5] , using cohomological method, showed that if a finite group G = T ✄< N is the semidirect product of a normal subgroup N and a subgroup T , then M (T ) is a direct factor of M (G) , where M (G) is the Schur-multiplicator of G and in the finite case , is the second cohomology group of G . In 1977 W.Haebich [1 Theorem 1.7]

On the Baer invariants of triples of groups

In this paper, we develop the theory of Baer invariants for triples of groups. First, we focus on the general properties of the Baer invariant of triples. Second, we prove that the Baer invariant of a triple preserves direct limits of directed systems of triples of groups. Moreover, we present a structure for the nilpotent multiplier of a triple of the free product in some cases. Finally, we give some conditions in which the Baer invariant of a triple is a torsion group.

On Baer Invariants of Triples of Groups

2011

Generalized Baer-Invariant of a Pair of Groups and Marginal Extension

2012

In this paper, we give connection between the order of the generalized Baerinvariant of a pair of finite groups and its factor groups, when ν is considered to be the specific variety. Moreover, we give a necessary and sufficient condition in which the generalized Baer-invariant of a pair of groups can be embedded into the generalized Baer-invariant of pair of its factor groups.

Subgroup Theorems for the Baer-Invariant of Groups

Journal of Algebra, 1998

M.R.Jones and J.Wiegold in have shown that if G is a finite group with a subgroup H of finite index n , then the n-th power of Schur multiplier of G , M (G) n , is isomorphic to a subgroup of M (H) .

Some Properties of Isologism of Groups and Baer-Invariants

Southeast Asian Bulletin of Mathematics, 2000

Let V be a variety of groups defined by the set of laws V . In this paper we study the concept of V-isologism of groups in terms of V-extensions and their connections with the Baer-invariant of groups are also discussed.

Some inequalities for the Baer-invariant of a pair of finite groups

Indagationes Mathematicae, 2007

In this paper we introduce the concept of Baer-invariant of a pair of groups with respect to a variety of groups v. Some inequalities for the Baer-invariant of a pair of finite groups are obtained, when v is considered to be the Schu~Baer variety. We also present a condition for which the order of the Baerinvariant of a pair of finite groups divides the order of the Baer-invariant of their factor groups. Finally, some inequalities for the Schur-multiplier of a pair of finite nilpotent groups and their factor groups are given. 1. INTRODUCTION Let F~ be the free group freely generated by a countable set and let V be a subset of F~. Let v be the variety of groups defined by the set of laws V. We assume that the reader is familiar with the notions of the verbal subgroup, V(G), and the marginal subgroup, V*(G), associated with the variety of groups v and a given group G (see also [11]). v is called a Schur-Baer variety if, for any group G for which the marginal factor group G~ V* (G) is finite, it follows that the verbal subgroup V (G) is also finite. I. Schur [13] proved that the variety of abelian groups is a Schur-Baer

On Baer Invariants of Groups with Topological Approach (original) (raw)

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