Some properties on the Schur multiplier of a pair of groups (original) (raw)

The Schur multiplier of central product of groups

Journal of Pure and Applied Algebra

Let G be a central product of two groups H and K. We study second cohomology group of G, having coefficients in a divisible abelian group D with trivial G-action, in terms of the second cohomology groups of certain quotients of H and K. In particular, for D = C * , some of our results provide a refinement of results from [Some groups with non-trivial multiplicators, Math. Z. 120 (1971), 307-308] and [On the Schur multiplicator of a central quotient of a direct product of groups, J. Pure Appl. Algebra 3 (1973), 73-82].

Relative Schur multipliers and universal extensions of group homomorphisms

2014

In this note, starting with any group homomorphism f : Γ → G, which is surjective upon abelianization, we construct a universal central extension u : U ։ G, under Γ with the same surjective property, such that for any central extension m : M ։ G, under f, there is a unique homomorphism U → M with the obvious commutation condition. The kernel of u is the relative Schur multiplier group H 2 (G, Γ; Z) as below. The case where G is perfect corresponds to Γ = 1. This yields homological obstructions to lifting solution of equations in G. Upon repetition, for finite groups, this gives a universal hypercentral factorization of the map f : Γ → G.

The partial Schur multiplier of a group

Journal of Algebra, 2013

Refining the technique worked out in previous papers, we give a characterization, up to equivalence, of the factor sets σ of partial projective representation of a group G over an algebraically closed field K in terms of equalities satisfied by σ. This allows one to conclude that any component of the partial Schur multiplier pM(G) is an epimorphic image of a direct power of K *. Moreover, it is shown that any component of pM(G) is an epimorphic image of the component pM G×G (G) of the equivalence classes of the totally defined partial factor sets. Examples with cyclic G are considered, in particular, the total component pM G×G (G) is determined when G is an arbitrary finite cyclic group. In this case pM G×G (G) is a direct power of K * .

On the Schur pair of groups

2020

In this paper, it is shown that $ (mathcal{V}, mathfrak{X}) $ is a Schur pair if and only if the Baer-invariant of an mathfrakXmathfrak{X}mathfrakX-group with respect to $ mathcal{V}$ is an mathfrakXmathfrak{X}mathfrakX-group. Also, it is proved that a locally mathfrakXmathfrak{X}mathfrakX class inherited the Schur pair property of , whenever mathfrakXmathfrak{X}mathfrakX is closed with respect to forming subgroup, images and extensions of its members. Subsequently, many interesting predicates about some generalizations of Schur's theorem and Schur multiplier of groups will be concluded.

On the Schur multiplier of finite 𝑝-groups of maximal class

Journal of Group Theory

In this article, we prove that the Schur multiplier of a finite 𝑝-group of maximal class of order p n p^{n} ( 4 ≤ n ≤ p + 1 4\leq n\leq p+1 ) is elementary abelian. The case n = p + 1 n=p+1 settles a question raised by Primož Moravec in an earlier article.

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).

On the Order of the Schur Multiplier of a Pair of Finite p-Groups II

Let G be a finite p-group and N be a normal subgroup of G with |N | = p n and |G/N | = p m . A result of shows that the order of the Schur multiplier of such a pair (G, N ) of finite pgroups is bounded by p 1 2 n(2m+n−1) and hence it is equal to p 1 2 n(2m+n−1)−t for some non-negative integer t. Recently, the authors have characterized the structure of (G, N ) when N has a complement in G and t ≤ 3. This paper is devoted to classification of pairs (G, N ) when N has a normal complement in G and t = 4, 5.

The Schur multiplier of groups of order~𝑝5

Journal of Group Theory

In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian p-groups of order {p^{5}} . As an application, we determine the capability of groups of order {p^{5}} .

On the Order of the Schur Multiplier of a Pair of Finite p-Groups

Journal of Advanced Research in Pure Mathematics, 2014

Let G be a finite p-group and N be a normal subgroup of G with |N | = p n and |G/N | = p m. A result of Ellis (1998) shows that the order of the Schur multiplier of such a pair (G, N) of finite pgroups is bounded by p 1 2 n(2m+n−1) and hence it is equal to p 1 2 n(2m+n−1)−t for some non-negative integer t. Recently, the authors have characterized the structure of (G, N) when N has a complement in G and t ≤ 3. This paper is devoted to classification of pairs (G, N) when N has a normal complement in G and t = 4, 5.

Schur multipliers of special 𝑝-groups of rank 2

Journal of Group Theory, 2020

Let G be a special p-group with center of order p 2. Berkovich and Janko asked to find the Schur multiplier of G in [1, Problem 2027]. In this article we answer this question by explicitly computing the Schur multiplier of these groups.