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On the faithful representations, of degree 2n, of certain extensions of 2-groups by orthogonal and symplectic groups

Part of: Linear algebraic groups and related topics Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

S. P. Glasby
S. P. Glasby
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia
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Abstract

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If R is a 2-group of symplectic type with exponent 4, then R is isomorphic to the extraspecial group , or to the central product 4 o 21+2n of a cyclic group of order 4 and an extraspecial group, with central subgroups of order 2 amalgamated. This paper gives an explicit description of a projective representation of the group A of automorphisms of R centralizing Z(R), obtained from a faithful representation of R of degree 2n. The 2-cocycle associated with this projective representation takes values which are powers of −1 if R is isomorphic to and powers of otherwise. This explicit description of a projective representation is useful for computing character values or computing with central extensions of A. Such central extensions arise naturally in Aschbacher's classification of the subgroups of classical groups.

MSC classification

Secondary: 20C15: Ordinary representations and characters 20G05: Representation theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 2 , April 1995 , pp. 232 - 247
Copyright
Copyright © Australian Mathematical Society 1995

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