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Finite Groups with All Maximal Subgroups of Prime or Prime Square Index

Published online by Cambridge University Press:  20 November 2018

Joseph Kohler
Joseph Kohler*
Affiliation:
California Institute of Technology Pasadena, California
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In this paper finite groups with the property M, that every maximal subgroup has prime or prime square index, are investigated. A short but ingenious argument was given by P. Hall which showed that such groups are solvable.

B. Huppert showed that a finite group with the property M, that every maximal subgroup has prime index, is supersolvable, i.e. the chief factors are of prime order. We prove here, as a corollary of a more precise result, that if G has property M and is of odd order, then the chief factors of G are of prime or prime square order. The even-order case is different. For every odd prime p and positive integer m we shall construct a group of order 2apb with property M which has a chief factor of order larger than m.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 435 - 442
Copyright
Copyright © Canadian Mathematical Society 1964

References

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