Finite p-groups having Schur multiplier of maximum order

[摘要] Let G be a non-abelian p-group of order p(n) and M(G) denote the Schur multiplier of G. Niroomand proved that vertical bar M(G)vertical bar <=. 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 vertical bar M(G)vertical bar attains this bound. In this article we show that there is no finite p-group G of nilpotency class c >= 3 for p not equal 3 such that vertical bar M(G)vertical bar attains this bound. Hence vertical bar M(G)vertical bar <= p(1/2(n+k-2)(n-k-1)) for p-groups G of class c >= 3 where p not equal 3. We also construct a p-group G for p = 3 such that vertical bar M(G)vertical bar attains the Niroomand's bound. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Schur multiplier;Finite p-group