Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T09:32:28.853Z Has data issue: false hasContentIssue false
  • English
  • Français

Conjugate p-Subgroups of Finite Groups

Published online by Cambridge University Press:  20 November 2018

John J. Currano
John J. Currano*
Affiliation:
Roosevelt University, Chicago, Illinois
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.

Throughout this paper, let p be a prime, P be a p-group of order pt , and ϕ be an isomorphism of a subgroup R of P of index p onto a subgroup Q which fixes no non-identity subgroup of P, setwise. In [2, Lemma 2.2], Glauberman shows that P can be embedded in a finite group G such that ϕ is effected by conjugation by some element g of G. We assume that P is thus embedded. Then Q = P ∩ Pg. Let H = 〈P,Pg〉 and V = [H,Z(Q)], so Q ⊲ H and V ⊲ H.

Let E(p) be the non-abelian group of order p3 which is generated by two elements of order p. Then E(p) is dihedral if p = 2 and has exponent p if p is odd. If p is odd, then E* (p) is defined in § 2 to be a particular group of order p6 and nilpotence class three.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 1 , February 1972 , pp. 17 - 28
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Currano, J., Finite p-groups with isomorphic subgroups (to appear).Google Scholar
2. Glauberman, G., Isomorphic subgroups of p-groups. I, Can. J. Math. 23 (1971), 983–1022.Google Scholar
3. Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
4. Kurosh, A. G., The theory of groups, second English edition, translated by Hirsh, K. A. (Chelsea, New York, 1960).Google Scholar