Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T11:00:38.878Z Has data issue: false hasContentIssue false
  • English
  • Français

A Characteristic Subgroup of a p-Stable Group

Published online by Cambridge University Press:  20 November 2018

George Glauberman
George Glauberman*
Affiliation:
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 let S be a Sylow p-subgroup of a finite group G. J. Thompson (13; 14) has introduced a characteristic subgroup JR(S) and has proved the following results:

(1.1) Suppose that p is odd. Then G has a normal p-complement if and only if C(Z(S)) and N(JR(S)) have normal p-complements.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1101 - 1135
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Alperin, J. and Gorenstein, D., Transfer and fusion infinite groups, J. Algebra 6 (1967), 242–255.Google Scholar
2. Artin, E., Geometric algebra (Interscience, New York, 1957).Google Scholar
3. Feit, W. and Thompson, J. G., Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029.Google Scholar
4. Glauberman, G., Weakly closed elements of Sylow subgroups (to appear in Math. Z.).Google Scholar
5. Gorenstein, D., p-constraint and the transitivity theorem, Arch. Math. 18 (1967), 355–358.Google Scholar
6. Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
7. Gorenstein, D. and Walter, J., On the maximal subgroups of finite simple groups, J. Algebra 1 (1964), 168-213.Google Scholar
8. Gorenstein, D. and Walter, J., The characterization of finite groups with dihedral Sylow 2-subgroups. I, J. Algebra 2 1965), 85–151.Google Scholar
9. Hall, M., The theory of groups (Macmillan, New York, 1959).Google Scholar
10. Hall, P. and Higman, G., On the p-length of p-solvable groups and reduction theorems for Burnsidës problem, Proc. London Math. Soc. (3) 6 (1956), 1.-42.Google Scholar
11. Higman, D. G., Focal series infinite groups, Can. J. Math. 5 (1953), 477–497.Google Scholar
12. Schenkman, E., Group theory (Van Nostrand, Princeton, N.J., 1965).Google Scholar
13. Thompson, J. G., Normal p-complements for finite groups, J. Algebra 1 (1964), 43–46.Google Scholar
14. Thompson, J. G., Factorizations for p-solvable groups, Pacific J. Math. 16 (1966), 371–372.Google Scholar
15. Thompson, J. G., A replacement theorem for p-groups and a conjecture (to appear in J. Algebra).Google Scholar
16. Zassenhaus, H., The theory of groups, 2nd ed. (Chelsea, New York, 1958).Google Scholar