# zbMATH — the first resource for mathematics

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.

##### MSC:
 20C34 Representations of sporadic groups 20D08 Simple groups: sporadic groups 20C40 Computational methods (representations of groups) (MSC2010)
GAP; nauty
Full Text:
##### References:
 [1] Breuer T., ”Potenzabbildungen, Untergruppenfusionen, Tafel-Automorphismen” (1991) [2] Brouwer A. E., Distance-Regular Graphs (1989) · Zbl 0747.05073 [3] Conway J. H., Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups (1985) · Zbl 0568.20001 [4] Conway J. H., Computers in Algebra pp 27– (1988) [5] Conway J. H., Groups, Combinatorics and Geometry pp 24– (1992) · doi:10.1017/CBO9780511629259.006 [6] Fischer B., ”Finite groups generated by 3-transpositions” (1969) · Zbl 0232.20040 [7] Isaacs I. M., Character Theory of Finite Groups (1976) · Zbl 0337.20005 [8] DOI: 10.1080/00927879508825406 · Zbl 0832.20025 · doi:10.1080/00927879508825406 [9] DOI: 10.1112/jlms/s2-39.1.89 · Zbl 0629.20006 · doi:10.1112/jlms/s2-39.1.89 [10] DOI: 10.1017/S0305004100067001 · Zbl 0622.20008 · doi:10.1017/S0305004100067001 [11] DOI: 10.1016/S0747-7171(08)80094-6 · Zbl 0789.20032 · doi:10.1016/S0747-7171(08)80094-6 [12] DOI: 10.1112/plms/s3-63.1.113 · Zbl 0695.20016 · doi:10.1112/plms/s3-63.1.113 [13] McKay B. D., Nauty User’s Guide (version 1.5) (1990) [14] Neubüser J., Computational Group Theory pp 195– (1984) [15] Praeger C. E., Low Rank Representations and Graphs for Sporadic Groups · Zbl 0860.20014 [16] Rowley P., ”On the Fi22-minimal parabolic geometry” (1993) [17] Schiffer U., ”Cliffordmatrizen” (1995) [18] Schönert M., GAP: Groups, Algorithms, and Programming, version 3 (1994) [19] Soicher L. H., Groups and Computation pp 287– (1993) [20] DOI: 10.1017/S0305004100061491 · Zbl 0551.20010 · doi:10.1017/S0305004100061491 [21] DOI: 10.1112/jlms/s2-32.3.460 · Zbl 0562.20006 · doi:10.1112/jlms/s2-32.3.460
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.