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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the existence of trivial intersection subgroups

Author: Mark P. Hale
Journal: Trans. Amer. Math. Soc. 157 (1971), 487-493
MSC: Primary 20.20
MathSciNet review: 0276315
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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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Keywords: Trivial intersection subgroup, Frobenius group, partitioned groups, Hall subgroup, centralizer of each of its nonidentity elements, exceptional characters
Article copyright: © Copyright 1971 American Mathematical Society

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