On the Order of the Schur Multiplier of a Pair of Finite p-Groups (original) (raw)

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.

Finite p -groups having Schur multiplier of maximum order

Journal of Algebra

Let G be a non-abelian p-group of order p n and M (G) denote the Schur multiplier of G. Niroomand proved that |M (G)| ≤ p 1 2 (n+k−2)(n−k−1)+1 for non-abelian p-groups G of order p n with derived subgroup of order p k. Recently Rai classified p-groups G of nilpotency class 2 for which |M (G)| attains this bound. In this article we show that there is no finite p-group G of nilpotency class c ≥ 3 for p = 3 such that |M (G)| attains this bound. Hence |M (G)| ≤ p 1 2 (n+k−2)(n−k−1) for p-groups G of class c ≥ 3 where p = 3. We also construct a p-group G for p = 3 such that |M (G)| attains the Niroomand's bound. 2 (n+k−2)(n−k−1)+1 we shall write |M (G)| attains the bound, throughout this paper. Recently Rai [16, Theorem 2.1] classified finite p-groups G of class 2 such that |M (G)| attains the bound. Aim of this paper is to continue this line of investigation and to look into the classification of arbitrary finite p-groups attending this bound. It, surprisingly turns out that for p = 3 there is no finite p-group G of nilpotency class c ≥ 3 such that |M (G)| attains the bound. Hence for p-groups G of class ≥ 3 and p = 3 we improve 2010 Mathematics Subject Classification. 20J99, 20D15.

On the Order of Schur Multipliers of Finite Abelian p-Groups

2010

Let GGG be a finite ppp-group of order pnp^{n}pn with ∣M(G)∣=pfracn(n−1)2−t,|M(G)|=p^{\frac{n(n-1)}{2}-t},∣M(G)∣=pfracn(n−1)2−t, where M(G)M(G)M(G) is the Schur multiplier of GGG. Ya.G. Berkovich, X. Zhou, and G. Ellis have determined the structure of GGG when t=0,1,2,3t=0,1,2,3t=0,1,2,3. In this paper, we are going to find some structures for an abelian ppp-group GGG with conditions on the exponents of G,M(G),G, M(G),G,M(G), and S_2M(G)S_2M(G)S_2M(G), where

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}} .

A Bound for the Exponent of the Schur Multiplier of some finite p-Groups

2000

Let G be a finite p-group of exponent pe. The paper is devoted to present a new bound for the exponent of the Schur multiplier of G, when G is of class 3, 4 or 5 and e satisfies in some conditions. 1991 Mathematics Subject Classification (Amer. Math. Soc.) ∗The first author’s research was partially supported by IPM.

The Schur multiplier of pairs of groups of order p3q

2016

Let (,) G N be a pair of groups in which N is a normal subgroup of. G Then, the Schur multiplier of pairs of groups (,), G N denoted by (,), M G N is an extension of the Schur multiplier of a group , G which is a functorial abelian group. In this research, the Schur multiplier of pairs of all groups of order 3 p q where p is an odd prime and p q is determined.

The Schur Multiplier of Pairs of Groups of Orderp2q

Mathematical Problems in Engineering

LetG,Nbe a pair of groups whereGis a group andNis a normal subgroup ofG. Then the Schur multiplier of pairs of groupsG,Nis a functorial abelian groupMG,N. In this paper,MG,Nfor groups of orderp2qwherepandqare prime numbers are determined.