Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T19:34:54.823Z Has data issue: false hasContentIssue false

Isomorphic Subgroups of Finite p-Groups. I

Published online by Cambridge University Press:  20 November 2018

George Glauberman*
Affiliation:
Department of Mathematics, The University of Chicago, 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.

Let p be a prime and P be a p-subgroup of a finite group G. Suppose that gG and that PPg has index p in P. In [4], we assumed that g normalizes no non-identity normal subgroup of P. We obtained some bounds on the order of P and some applications to the case in which p = 2 and P is a Sylow 2-subgroup of 〈P, Pg〉. In this paper, we examine this situation further by considering the isomorphism ϕ of PPg-l onto PPg given by ϕ(x) = xg. We actually consider arbitrary isomorphisms ϕ between two subgroups of index p in P. However, an easy argument (Lemma 2.3) shows that every such ϕ can be obtained as above for some G and some g. We obtain some results concerning the nilpotence class rather than the order of P.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Bruck, R. H., A survey of binary systems (Springer, Berlin, 1958).Google Scholar
2. Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups (Springer, Berlin, 1957).Google Scholar
3. Currano, J., Conjugate p-subgroups with maximal intersection, Ph.D. Thesis, University of Chicago, 1970.Google Scholar
4. Glauberman, G., Normalizers of p-subgroups infinite groups, Pacific J. Math. 29 (1969), 137144.Google Scholar
5. Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
6. Hall, M., The theory of groups (Macmillan, New York, 1959).Google Scholar
7. Huppert, B., Endliche Gruppen. I (Springer, Berlin, 1967).Google Scholar
8. Kurosh, A. G., The theory of groups, second English edition, translated by Hirsch, K. A. (Chelsea, New York, 1960).Google Scholar
9. Passman, D. S., Permutation groups (Benjamin, New York, 1968).Google Scholar
10. Sims, C. C., Graphs and finite permutation groups, Math. Z. 95 (1967), 7686.Google Scholar