Finite p-groups having Schur multiplier of maximum order

2016/10/22 by Hatui, Sumana
#20D15 #20J99 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1610.07042

Abstract

Let G be a non-abelian p-group of order pⁿ 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 pⁿ 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.

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