Remarks on proficient groups. Dedicated to the memory of Karl Gruenberg
Remarks on proficient groups. Dedicated to the memory of Karl Gruenberg
If a finite group G has a presentation with d generators and r relations, it is well-known that r ? d is at least the rank of the Schur multiplier of G; a presentation is called efficient if equality holds. There is an analogous definition for proficient profinite presentations. We show that many perfect groups have proficient presentations. Moreover, we prove that infinitely many alternating groups, symmetric groups and their double covers have proficient presentations.
presentations of finite groups, proficient groups, efficient groups, profinite groups, cohomology
Guralnik, R.M.
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Kantor, W.M.
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Kassabov, Martin
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Lubotzky, A.
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Guralnik, R.M.
715333e2-0a50-4960-943b-2319c9cad09a
Kantor, W.M.
40ce4091-5d7b-4188-8f80-81734c071a25
Kassabov, Martin
b78efbac-c468-4838-ac56-5f23181f595c
Lubotzky, A.
b89c8821-4d60-428e-a84e-ec923d2396f5
Guralnik, R.M., Kantor, W.M., Kassabov, Martin and Lubotzky, A.
(2009)
Remarks on proficient groups. Dedicated to the memory of Karl Gruenberg.
Journal of Algebra.
(In Press)
Abstract
If a finite group G has a presentation with d generators and r relations, it is well-known that r ? d is at least the rank of the Schur multiplier of G; a presentation is called efficient if equality holds. There is an analogous definition for proficient profinite presentations. We show that many perfect groups have proficient presentations. Moreover, we prove that infinitely many alternating groups, symmetric groups and their double covers have proficient presentations.
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Accepted/In Press date: May 2009
Keywords:
presentations of finite groups, proficient groups, efficient groups, profinite groups, cohomology
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Local EPrints ID: 155187
URI: http://eprints.soton.ac.uk/id/eprint/155187
ISSN: 0021-8693
PURE UUID: 4914f08f-0a51-4045-86d0-9029aa98d646
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Date deposited: 28 May 2010 11:35
Last modified: 14 Mar 2024 01:37
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Contributors
Author:
R.M. Guralnik
Author:
W.M. Kantor
Author:
Martin Kassabov
Author:
A. Lubotzky
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