On Sylow 2-subgroups with no normal Abelian subgroups of rank 3, in finite fusion-simple groups

Patterson, Anne R.

doi:10.1090/S0002-9947-1974-0342608-7

On Sylow $2$-subgroups with no normal Abelian subgroups of rank $3$, in finite fusion-simple groups
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by Anne R. Patterson PDF

Trans. Amer. Math. Soc. 187 (1974), 1-67 Request permission

Abstract:

Let T be any finite 2-group which has a normal four-group but has no normal Abelian subgroup of rank 3, and assume T is not the dihedral group of order 8. If T is a Sylow 2-subgroup of a finite fusion-simple group G, it follows (Thompson) from Glauberman’s ${Z^ \ast }$-theorem that T has exactly one normal four-group, say W. This paper establishes what isomorphism types of T can so occur under the hypothesis that ${{\mathbf {N}}_G}(T) = T{{\mathbf {C}}_G}(T)$ and the three nonidentity elements of W are not all G-conjugate. All T arrived at in this paper are known to so occur. The reason for this hypothesis is that the similar situation for T with a normal four-group and no normal Abelian subgroup of rank 3, where T is a Sylow 2-subgroup of a finite simple group G but without the above hypothesis, had been analyzed earlier by the author (under her maiden name, MacWilliams; Trans. Amer. Math. Soc. 150 (1970), 345-408).

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