A note on p-central groups
(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.
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/6d980302-5492-4bd6-8cb5-57ad8b6a15a1
- author
- Camina, Rachel and Thillaisundaram, Anitha LU
- publishing date
- 2013
- 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
- 2024-08-14 14:19:39
@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}}, }