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Existence of finite groups with classical commutator subgroup

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  09 April 2009

Michael D. Miller
Michael D. Miller
Affiliation:
Department of Mathematics University of California, Los Angeles California 90024, USA
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Abstract

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Given a group G, we may ask whether it is the commutator subgroup of some group G. For example, every abelian group G is the commutator subgroup of a semi-direct product of G x G by a cyclic group of order 2. On the other hand, no symmetric group Sn(n>2) is the commutator subgroup of any group G. In this paper we examine the classical linear groups over finite fields K of characteristic not equal to 2, and determine which can be commutator subgroups of other groups. In particular, we settle the question for all normal subgroups of the general linear groups GLn(K), the unitary groups Un(K) (n≠4), and the orthogonal groups On(K) (n≧7).

MSC classification

Secondary: 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 1 , February 1978 , pp. 41 - 44
Copyright
Copyright © Australian Mathematical Society 1978

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