Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T15:19:52.005Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite equilibrated groups

Published online by Cambridge University Press:  24 October 2008

Norman Blackburn ,
Marian Deaconescu  and
Avinoam Mann
Norman Blackburn
Affiliation:
Department of Mathematics, University of Manchester, Manchester, M13 9PL
Marian Deaconescu
Affiliation:
Faculty of Mathematics, University of Timisoara, Timisoara, Rumania
Avinoam Mann
Affiliation:
Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
Get access Rights & Permissions [Opens in a new window]

Extract

If H, K are subgroups of a group G, then HK is a subgroup of G if and only if HK = KH. This condition certainly holds if H ≤ NG(K) or K ≤ NG(H). But the majority of groups can also be expressed as HK, where neither H nor K is normal. In this paper we consider groups G for which no subgroup G1 can be expressed as the product of non-normal subgroups of G1. Such a group is said to be equilibrated. Thus G is equilibrated if and only if either H ≤ NG(K) or K ≤ NG(H) whenever H, K and HK are subgroups of G.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 4 , November 1996 , pp. 579 - 588
Copyright
Copyright © Cambridge Philosophical Society 1996

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]Blackburn, N.On a special class of p-groups. Acta Math. 100, 45–92 (1958).CrossRefGoogle Scholar
[2]Blackburn, N.On prime-power groups with two generators. Proc. Cambridge Phil. Soc. 54, 327–337 (1958).CrossRefGoogle Scholar
[3]Gaschütz, W.Über die Φ-Untergruppe endlicher Gruppen. Math. Z. 58, 160–170 (1953).CrossRefGoogle Scholar
[4]Glauberman, G.Factorizations in local subgroups of finite groups. Regional Conf. series in Math. 33 (1977), A.M.S., Providence, R.I.Google Scholar
[5]Gorenstein, D.Finite groups (Harper and Row, 1968).Google Scholar
[6]Huppert, B.Endliche Gruppen I (Springer-Verlag, 1967).CrossRefGoogle Scholar
[7]Huppert, B. and Blackburn, N.Finite Groups II (Springer-Verlag, 1982).CrossRefGoogle Scholar
[8]Huppert, B. and Blackburn, N.Finite Groups II (Springer-Verlag, 1982).CrossRefGoogle Scholar
[9]Ito, N.On the factorizations of the linear fractional groups LF(2, pn). Acta Sci. Math. Szeged 15, 79–84 (1953).Google Scholar
[10]Janko, Z.A new finite simple group with Abelian Sylow 2-subgroups and its characterization. J. of Algebra, 3, 147–186 (1966).CrossRefGoogle Scholar