Sp. Glasby, ON THE FAITHFUL REPRESENTATIONS, OF DEGREE 2(N), OF CERTAIN EXTENSIONS OF 2-GROUPS BY ORTHOGONAL AND SYMPLECTIC GROUPS, Journal of the Australian Mathematical Society. Series A. Pure mathematics and statistics, 58, 1995, pp. 232-247
Citations number
12
Categorie Soggetti
Mathematics, General","Statistic & Probability",Mathematics,"Statistic & Probability
If R is a 2-group of symplectic type with exponent 4, then R is isomor
phic to the extraspecial group 2(epsilon)(1+2n), or to the central pro
duct 4 o 2(1+2n) of a cyclic group of order 4 and an extraspecial grou
p, 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 represe
ntation of R of degree 2(n). The 2-cocycle associated with this projec
tive representation takes values which are powers of -1 if R is isomor
phic to 2(epsilon)(1+2n) and powers of root-1 otherwise. This explicit
description of a projective representation is useful for computing ch
aracter values or computing with central extensions of A. Such central
extensions arise naturally in Aschbacher's classification of the subg
roups of classical groups.