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The existence of subgroups of given order in finite groups

Published online by Cambridge University Press:  24 October 2008

D. H. McLain
D. H. McLain
Affiliation:
Mathematics DepartmentGlasgow University
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Extract

1. Discussion of results

1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, (G), of G. More generally, if H and K are subgroups of G, and H ≥ K, then (G:K) = (G:H)(H:K), where (G:K) denotes the index of K in G, etc. We call a number a possible order of a subgroup of G if it is a divisor of (G), and a possible order of a subgroup of G containing a subgroup H if it is a divisor of (G) and a multiple of (H). In this paper we discuss conditions on G for the existence of subgroups of every possible order, the existence of subgroups of every possible order containing arbitrary subgroups, and similar properties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 2 , April 1957 , pp. 278 - 285
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

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