The multiplicity free permutation representations of the Ree groups (2)G(2)(q), the Suzuki groups B-2(2)(q), and their automorphism groups

摘要

Let G a simple group of type B-2(2)(q) or (2)G(2)(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A less than or equal to Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = B-2(2)(q) we show that there is only one non-trivial multiplicity free permutation representation, namely 2 the representation of G associated to the action on GIB. If G = (2)G(2)(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on GIB and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where His isomorphic to a subgroup containing B if G = B-2(2)(q), and containing M if G = (2)G(2)(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs. [References: 19]