Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T15:00:10.816Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Burnside, W.Theory of groups of finite order, 2nd ed. (Cambridge, 1911).Google Scholar
(2)Hall, P.A note on soluble groups. J. Lond. Math. Soc. 3 (1928), 98–105.CrossRefGoogle Scholar
(3)Hall, P.A characteristic property of soluble groups. J. Lond. Math. Soc. 12 (1937), 198–200.CrossRefGoogle Scholar
(4)Hall, P.On the Sylow systems of soluble groups. Proc. Lond. Math. Soc. (2) 43 (1937), 316–23.Google Scholar
(5)Huppert, B.Normalteiler und maximale Untergruppen endlicher Gruppen. Math. Z. 60 (1954), 409–34.CrossRefGoogle Scholar
(6)Ore, O.Contributions to the theory of groups of finite order. Duke Math. J. 5 (1939), 431–60.CrossRefGoogle Scholar
(7)Zappa, G.A remark on a recent paper of O. Ore. Duke Math. J. 6 (1940), 511–12.Google Scholar