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

Author:
Anne R. Patterson

Journal:
Trans. Amer. Math. Soc. **187** (1974), 1-67

MSC:
Primary 20D20

MathSciNet review:
0342608

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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 -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 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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DOI:
https://doi.org/10.1090/S0002-9947-1974-0342608-7

Keywords:
2-group,
Sylow 2-subgroup,
rank,
fusion-simple,
conjugate,
transfer homomorphism,
fusion

Article copyright:
© Copyright 1974
American Mathematical Society