On the existence of trivial intersection subgroups

Hale, Mark P.

doi:10.1090/S0002-9947-1971-0276315-3

On the existence of trivial intersection subgroups
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by Mark P. Hale PDF

Trans. Amer. Math. Soc. 157 (1971), 487-493 Request permission

Abstract:

Let $G$ be a transitive nonregular permutation group acting on a set $X$, and let $H$ be the subgroup of $G$ fixing some element of $X$. Suppose each nonidentity element of $H$ fixes exactly $b$ elements of $X$. If $b = 1,G$ is a Frobenius group, and it is well known that $H$ has only trivial intersection with its conjugates. If $b > 1$, it is shown that this conclusion still holds, provided $H$ satisfies certain natural conditions. Applications to the study of Hall subgroups and certain simple groups related to Zassenhaus groups are given.

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