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Characterizations of the Fischer groups. I, II, III


Author: David Parrott
Journal: Trans. Amer. Math. Soc. 265 (1981), 303-347
MSC: Primary 20D05
DOI: https://doi.org/10.1090/S0002-9947-1981-0610952-5
MathSciNet review: 610952
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Abstract: 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)$.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0610952-5
Article copyright: © Copyright 1981 American Mathematical Society

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