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ON CENTRALISERS AND NORMALISERS FOR GROUPS

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  02 August 2012

BORIS ŠIROLA
BORIS ŠIROLA*
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia (email: [email protected])
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Abstract

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Let 𝕂 be a field, char(𝕂)≠2, and G a subgroup of GL(n,𝕂). Suppose g↦g♯ is a 𝕂-linear antiautomorphism of G, and then define G1={g∈G∣g♯g=I}. For C being the centraliser 𝒞G (G1) , or any subgroup of the centre 𝒵(G) , define G(C) ={g∈G∣g♯g∈C}. We show that G(C) is a subgroup of G, and study its structure. When C=𝒞G (G1) , we have that G(C) =𝒩G (G1) , the normaliser of G1 in G. Suppose 𝕂 is algebraically closed, 𝒞G (G1) consists of scalar matrices and G1 is a connected subgroup of an affine group G. Under the latter assumptions, 𝒩G (G1) is a self-normalising subgroup of G. This holds for a number of interesting pairs (G,G1); in particular, for those that we call parabolic pairs. As well, for a certain specific setting we generalise a standard result about centres of Borel subgroups.

Keywords

centralisernormaliserself-normalising subgroupparabolic subgroup

MSC classification

Secondary: 20E34: General structure theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 481 - 494
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

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