A note on the order of the Schur multiplier of p-groups

2016/06/05 by Pradeep K. Rai, Rai, Pradeep K.
Mathematics · Engineering · #Finite Group Theory Research #graph theory and CDMA systems #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1606.01493

Abstract

Let G be a finite p-group of order pⁿ with |G'| = p^k. Let M(G) denotes the Schur multiplier of G. A classical result of Green states that |M(G)| ≤ p^{(1)/(2)n(n-1)}. In 2009, Niroomand, improving Green's and other bounds on |M(G)| for a non-abelain p-group G, proved that |M(G)| ≤ p^{(1)/(2)(n-k-1)(n+k-2)+1}. In this article we note that a bound, obtained earlier, by Ellis and Weigold is more general than the bound of Niroomand. We derive from the bound of Ellis and Weigold that |M(G)| ≤ p^{(1)/(2)(d(G)-1)(n+k-2)+1} for a non-abelain p-group G. Moreover, we sharpen the bound of Ellis and Weigold and as a consequence derive that if G^ab is not homocyclic then |M(G)| ≤ p^{(1)/(2)(d(G)-1)(n+k-3)+1}. We further note an improvement in an old bound given by Vermani. Finally we note, for a p-group of coclass r, that |M(G)| ≤ p^{(1)/(2)(r2-r)+kr+1}. This improves a bound by Moravec.

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