The primitive distance-transitive representations of the Fischer groups. (English) Zbl 0855.20014

A permutation representation of a group \(G\) is a distance-transitive representation (DTR) if a \(G\)-invariant graph \(\Gamma\) can be defined on the points, such that \(G\) acts transitively on pairs of points at any given distance in \(\Gamma\).
This paper describes in some detail the computer-assisted classification of all primitive DTRs of Fischer’s three sporadic simple groups and their automorphism groups. This forms part of the complete determination of such representations for all the sporadic simple groups and their automorphism groups by A. A. Ivanov, J. Saxl and the authors [Commun. Algebra 23, No. 9, 3379-3427 (1995; Zbl 0832.20025)].
The method relies heavily on much previously computed information about these groups, such as their character tables, maximal subgroups, and the character tables of their maximal subgroups (when available). The first step is to determine the primitive multiplicity-free permutation characters. (Note that all multiplicity-free permutation characters of the sporadic simple groups and their automorphism groups are given in a later paper by T. Breuer and K. Lux [Commun. Algebra 24, No. 7, 2293-2316 (1996; see the preceding review Zbl 0855.20013)].) The corresponding permutation representations are then usually constructed more or less explicitly and analysed with various tools to see whether they are distance transitive.
It turns out that the only DTRs of the Fischer groups are (a) the actions on the 3-transpositions, and (b) the action of \(\text{Fi}_{22}\) on the cosets of \(O_7(3)\), and the action of \(\text{Fi}_{23}\) on the cosets of \(O^+_8(3):S_3\). The paper contains a wealth of other useful information about permutation representations of the Fischer groups.


20C34 Representations of sporadic groups
20D08 Simple groups: sporadic groups
20C40 Computational methods (representations of groups) (MSC2010)


GAP; nauty
Full Text: DOI Euclid EuDML EMIS


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