A NOTE ON p-CENTRAL GROUPS

作者：Camina Rachel^*; Thillaisundaram Anitha

来源：Glasgow Mathematical Journal, 2013, 55(2): 449-456.

DOI：10.1017/S0017089512000687

A group G is n-central if G(n) <= Z(G), that is the subgroup of G generated by n-powers of G lies in the centre of G. We investigate p(k)-central groups for p a prime number. For G a finite group of exponent p(k), the covering group of G is p(k)-central. Using this we show that the exponent of the Schur multiplier of G is bounded by p(inverted right perpendicularc/p-1inverted left perpendicular), where c is the nilpotency class of G. Next we give an explicit bound for the order of a finite p(k)-central p-group of coclass r. Lastly, we establish that for G, a finite p-central p-group, and N, a proper non-maximal normal subgroup of G, the Tate cohomology H-n(G/N, Z(N)) is non-trivial for all n. This final statement answers a question of Schmid concerning groups with non-trivial Tate cohomology.

• 出版日期2013-5