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A NOTE ON p-CENTRAL GROUPS

Published online by Cambridge University Press:  25 February 2013

RACHEL CAMINA  and
ANITHA THILLAISUNDARAM
RACHEL CAMINA
Affiliation:
Fitzwilliam College, Cambridge, CB3 0DG, UK e-mail: rdc26@dpmms.cam.ac.uk
ANITHA THILLAISUNDARAM
Affiliation:
Magdalene College, Cambridge, CB3 0AG, UK e-mail: anitha.t@cantab.net
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Abstract

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A group G is n-central if Gn ≤ Z(G), that is the subgroup of G generated by n-powers of G lies in the centre of G. We investigate pk-central groups for p a prime number. For G a finite group of exponent pk, the covering group of G is pk-central. Using this we show that the exponent of the Schur multiplier of G is bounded by $p^{\lceil \frac{c}{p-1} \rceil}$, where c is the nilpotency class of G. Next we give an explicit bound for the order of a finite pk-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 Hn(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.

Keywords

20D15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 2 , May 2013 , pp. 449 - 456
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013

References

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