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A note on p-central groups

Camina, Rachel and Thillaisundaram, Anitha LU (2013) In Glasgow Mathematical Journal 55(2). p.449-456
Abstract
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 p k -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⌈c/p-1⌉, 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.
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author
and
publishing date
type
Contribution to journal
publication status
published
subject
in
Glasgow Mathematical Journal
volume
55
issue
2
pages
449 - 456
publisher
Cambridge University Press
external identifiers
  • scopus:84875998492
ISSN
1469-509X
DOI
10.1017/S0017089512000687
language
English
LU publication?
no
id
6d980302-5492-4bd6-8cb5-57ad8b6a15a1
date added to LUP
2024-06-07 14:34:05
date last changed
2025-04-04 15:20:35
@article{6d980302-5492-4bd6-8cb5-57ad8b6a15a1,
  abstract     = {{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 p k -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⌈c/p-1⌉, 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.}},
  author       = {{Camina, Rachel and Thillaisundaram, Anitha}},
  issn         = {{1469-509X}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{449--456}},
  publisher    = {{Cambridge University Press}},
  series       = {{Glasgow Mathematical Journal}},
  title        = {{A note on p-central groups}},
  url          = {{http://dx.doi.org/10.1017/S0017089512000687}},
  doi          = {{10.1017/S0017089512000687}},
  volume       = {{55}},
  year         = {{2013}},
}