Characterizations of the Fischer groups. I, II, III

Abstract

B. Fischer, in his work on finite groups which contain a conjugacy class of 3 3 -transpositions, discovered three new sporadic finite simple groups, usually denoted M ( 22 ) M(22) , M ( 23 ) M(23) and M ( 24 ) ′ M(24)’ . In Part I two of these groups, M ( 22 ) M(22) and M ( 23 ) M(23) , are characterized by the structure of the centralizer of a central involution. In addition, the simple groups U 6 ( 2 ) {U_6}(2) (often denoted by M ( 21 ) ) M(21)) and P Ω ( 7 , 3 ) P\Omega (7,3) , both of which are closely connected with Fischer’s groups, are characterized by the same method. The largest of the three Fischer groups M ( 24 ) M(24) is not simple but contains a simple subgroup M ( 24 ) ′ M(24)’ of index two. In Part II we give a similar characterization by the centralizer of a central involution of M ( 24 ) M(24) and also a partial characterization of the simple group M ( 24 ) ′ M(24)’ . The purpose of Part III is to complete the characterization of M ( 24 ) ′ M(24)’ by showing that our abstract group G G is isomorphic to M ( 24 ) ′ M(24)’ . We first prove that G G contains a subgroup X ≅ M ( 23 ) X \cong M(23) and then we construct a graph (on the cosets of X X ) which is shown to be isomorphic to the graph for M ( 24 ) M(24) .