On 2-groups with no normal abelian subgroups of rank 3, and their occurrence as Sylow 2-subgroups of finite simple groups

MacWilliams, Anne R.

doi:10.1090/S0002-9947-1970-0276324-3

On $2$-groups with no normal abelian subgroups of rank $3$, and their occurrence as Sylow $2$-subgroups of finite simple groups
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by Anne R. MacWilliams

Trans. Amer. Math. Soc. 150 (1970), 345-408

DOI: https://doi.org/10.1090/S0002-9947-1970-0276324-3

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Abstract:

We prove that in a finite $2$-group with no normal Abelian subgroup of rank $\geqq 3$, every subgroup can be generated by four elements. This result is then used to determine which $2$-groups $T$ with no normal Abelian subgroup of rank $\geqq 3$ can occur as ${S_2}$’s of finite simple groups $G$, under certain assumptions on the embedding of $T$ in $G$.

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