Characterizations of the Fischer Groups. I, II, III

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