Recent Progress on Varietal Schur Multipliers of some Products of Groups (original) (raw)
Related papers
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.
On Nilpotent Multipliers of Some Verbal Products of Groups
2010
The paper is devoted to finding a homomorphic image for the ccc-nilpotent multiplier of the verbal product of a family of groups with respect to a variety mathcalV{\mathcal V}mathcalV when mathcalVsubseteqmathcalNc{\mathcal V} \subseteq {\mathcal N}_{c}mathcalVsubseteqmathcalNc or mathcalNcsubseteqmathcalV{\mathcal N}_{c}\subseteq {\mathcal V}mathcalNcsubseteqmathcalV. Also a structure of the ccc-nilpotent multiplier of a special case of the verbal product, the nilpotent product, of cyclic
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 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
Polynilpotent multipliers of finitely generated abelian groups
In this paper, we present an explicit formula for the Baer invariant of a finitely generated abelian group with respect to the variety of polynilpotent groups of class row (c 1 ,. .. , c t), N c1,...,ct. In particular, one can obtain an explicit structure of the ℓ-solvable multiplier (the Baer invariant with respect to the vaiety of solvable groups of length at most ℓ ≥ 1, S ℓ .) of a finitely generated abelian group.
The Baer Invariant of Semidirect and Verbal Wreath Products of Groups
2011
W. Haebich (1977, Journal of Algebra {\bf 44}, 420-433) presented some formulas for the Schur multiplier of a semidirect product and also a verbal wreath product of two groups. The author (1997, Indag. Math., (N.S.), {\bf 8}({\bf 4}), 529-535) generalized a theorem of W. Haebich to the Baer invariant of a semidirect product of two groups with respect to the
The Higher Schur-Multiplicator of Certain Classes of Groups
Arxiv preprint arXiv: …, 2011
The paper is devoted to calculating the higher Schur-multiplicator of certain classes of groups with respect to the variety of nilpotent groups. Our results somehow generalize the works of M.R.R. , and N.D. Gupta and M.R.R. Moghaddam (1993).
Journal of the Australian Mathematical Society, 1981
In 19S7 P. Hall conjectured that every (finitely based) variety has the property that, for every group G, if the marginal factor-group is finite, then the verbal subgroup is also finite. The content of this paper is to present a precise bound for the order of the verbal subgroup of a group G when the marginal factor-group is of order p" (p a prime and n > 1) with respect to the variety of polynilpotent groups of a given class row. We also construct an example to show that the bound is attained and furthermore, we obtain a bound for the order of the Baer-invariant of a finite /"-group with respect to the variety of polynilpotent 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]
Higher Schur-multiplicator of a Finite Abelian Group
Arxiv preprint arXiv:1104.0400, 2011
In this paper we obtain an explicit formula for the higher Schurmultiplicator of an arbitrary finite abelian group with respect to the variety of nilpotent groups of class at most c ≥ 1 . 1991 Mathematics Subject Classification: Primary 20F12,20F18,20K25