Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T18:25:51.792Z Has data issue: false hasContentIssue false
  • English
  • Français

On the other pαqβ theorem of Burnside

Published online by Cambridge University Press:  20 January 2009

Arie Bialostocki
Arie Bialostocki
Affiliation:
Department of Mathematics and Statistics, University of Idaho, Moscow, Idaho, 83843, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The “other” pαqβ theorem of Burnside states the following:

Theorem A.l. Let G be a group of order pαqβ, where p and q are distinct primes. If pα>qβ, then Op(G)≠1 unless

(a) p is a Mersenne prime and q = 2;

(b) p = 2 and q is a Fermat prime; or

(c) p = 2 and q = 7.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 1 , February 1987 , pp. 41 - 49
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1.Bialostocki, A., On products of two nilpotent subgroups of a finite group, Israel J. Math. 20 (1975), 178–188.CrossRefGoogle Scholar
2.Bialostocki, A., The nilpotency class of the pSylow subgroups of GL(n, q) where (p, q) = l, Canad. Math. Bull. 29 (2) (1986), 218–223.CrossRefGoogle Scholar
3.Burnside, W., On groups of Order p αq β II, Proc. London Math. Soc. 2 (1904), 432–437.Google Scholar
4.Carter, R. and Fong, P., The Sylow 2-subgroups of the finite classical groups, J. Algebra 1 (1964), 139–151.CrossRefGoogle Scholar
5.Coates, M., Dwan, M. and Rose, J., A note on Burnside's other p xq B theorem, J. London Math. Soc. (2) 14 (1976), 160–166.CrossRefGoogle Scholar
6.Glauberman, G., On Burnside's other p xq B theorem, Pacific J. Math. 56 (1975), 469–475.CrossRefGoogle Scholar
7.Gorenstein, D., Finite Groups (Harper and Row, New York, 1968).Google Scholar
8.Monakhov, V. S., Order of Sylow subgroups of the general linear group, Algebra i Logika 17 (1978), 79–85.Google Scholar
9.Monakhov, V. S., Invariant subgroups of biprimary groups, Mat. Zametki 18 (1975), 877–886.Google Scholar
10.Passman, D. S., Permutation Groups (W. A. Benjamin Inc., New York, Amsterdam, 1968).Google Scholar
11.Weir, A., Sylow p-subgroups of the classical groups over finite fields with characteristic prime to p, Proc. Amer. Math. Soc. 6 (1955), 529–533.Google Scholar
12.Wolf, T. R., Solvable and nilpotent subgroups of GL(n, q m), Canad. J. Math. 24 (1982), 1097–1111.CrossRefGoogle Scholar