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CLASSICAL GROUPS IN DIMENSION 3 AS COMPLETIONS OF THE GOLDSCHMIDT G3-AMALGAM

Published online by Cambridge University Press:  13 February 2001

CHRISTOPHER PARKER  and
PETER ROWLEY
CHRISTOPHER PARKER
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT
PETER ROWLEY
Affiliation:
Department of Mathematics, University of Manchester Institute of Science and Technology, PO Box 88, Manchester M60 1QD
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Abstract

The aim of this paper is to determine which (finite) 3-dimensional classical groups are completions of the Goldschmidt G3-amalgam. We recall, first, that an amalgam (of rank 2) consists of three groups P1, P2, B and two group monomorphisms ϕ1, ϕ2 such that

formula here

Usually, when ϕ1 and ϕ2 are understood, this amalgam is denoted by [Ascr ](P1, P2, B). Now a group G is a completion of [Ascr ](P1, P2, B) if there exist group homomorphisms ψi[ratio ]Pi → G (i = 1, 2) satisfying G = 〈im ψ1, im ψ2〉 and ψ1ϕ1 = ψ2ϕ2[ratio ]B→G. The Goldschmidt G3-amalgam, which appears in [6], is defined as follows: P1 ≅ Sym(4) ≅ P2, B ≅ Dih(8) (Dih(n) denotes the dihedral group of order n) with ϕ−11(O2(P1)) ≠ ϕ−12(O2(P2)).

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 62 , Issue 3 , December 2000 , pp. 802 - 812
Copyright
The London Mathematical Society 2000

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