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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
Affiliation:
Mathematics DepartmentGlasgow University

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 HK, 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
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

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