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A Note on Supersoluble Groups

Published online by Cambridge University Press:  20 November 2018

D. L. Johnson
D. L. Johnson*
Affiliation:
University of Illinois at Urbana-Champaign, Urbana, Illinois
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Let H be a subgroup of a group G (all groups considered throughout this article are finite); then H will be called primitive if the subgroup

is distinct from H. Such subgroups, which are also called meet-irreducible, arise naturally in connection with minimal permutation representations of groups and in other contexts; for example, every subgroup of a group G can be written as an intersection of primitive subgroups of G, and the set of all primitive subgroups of G is characterized by its minimality with respect to this property. While maximal subgroups are always primitive, most groups contain non-maximal subgroups which are primitive (see remark at end of article). Note that a subgroup H of an abelian group G is primitive if, and only if, G/H is cyclic of prime-power order.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 562 - 564
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Huppert, B., Endliche Gruppen, Vol. I (Springer-Verlag, Berlin, 1967).Google Scholar