Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-20T02:46:12.030Z Has data issue: false hasContentIssue false
  • English
  • Français

The order of groups satisfying a converse to Lagrange's theorem

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  26 February 2010

Naihuan Jing
Naihuan Jing
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, U.S.A. E-mail: jing@math.ncsu.edu
Get access

Abstract

One of the converse statements to Lagrange's theorem is that, for each subgroup H of G and any prime factor p of |G: H|, there exists a subgroup K such that H≤K≤G with |K: H | = p. This paper treats integers n such that all groups of order n have this property.

MSC classification

Secondary: 20F16: Solvable groups, supersolvable groups
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 197 - 204
Copyright
Copyright © University College London 2000

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

[Bl]Bechtell, H.. Theory of Groups (Addison-Wesley, 1971).Google Scholar
[B2]Berger, T. R.. A converse to Lagrange's theorem. J. Austral. Math. Soc., 25 (1978), 291–313.CrossRefGoogle Scholar
[BW]Bray, H. G., Deskins, W. E., Johnson, D., Humphreys, J. F., Puttaswamaiah, M., Venzke, P., Walls, G. L. and Weinstein, M.. Between nilpotent and solvable (Polygonal Publishing House, Passaic, 1982).Google Scholar
[FZ]Fan, Y. and Zhang, Y. D.. On supersolubility of groups of order n. Jour. Math. (Wuhan), 1 (1981), 86–95.Google Scholar
[H]Huppert, B.. Endliche Gruppen I (Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[Ml]McCarthy, D. J.. A survey of partial converses to Lagrange's theorem on finite groups. Trans. New York Acad. Sci. (2), 33 (1971), 586–594.CrossRefGoogle Scholar
[M2]McLain, D. H.. The existence of subgroups of given order in finite groups. Proc. Cambridge Philos. Soc., 53 (1957), 278–285.CrossRefGoogle Scholar
[PI]Pazderski, G.. Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehören. Arch. Math., 10 (1959), 331–343.CrossRefGoogle Scholar
[P2]Pazderski, G.. The orders of which only belong to metabelian groups. Math. Nachr., 95 (1980), 7–16.CrossRefGoogle Scholar
[SI]Struik, R. R.. Partial converses to Lagrange's theorem. Commun. Alg. 6(5) (1978), 421–482.CrossRefGoogle Scholar
[S2]Struik, R. R.. Partial converses to Lagrange's theorem II. Commun. Alg., 9(1) (1981), 1–22.CrossRefGoogle Scholar
[Z]Zhang, Y. D.. An outline of supersolvable groups. Adv. in Math. (Beijing), 11 (1982), 200–205.Google Scholar