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Elementary abelian operator groups and admissible formations

Part of: Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Fletcher Gross
Fletcher Gross
Affiliation:
Department of Pure Mathematics Australian National UniversityCanberra ACT 2600, Australia Department of Mathematics University of UtahSalt Lake City Utah 84112, U.S.A.
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Abstract

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Suppose the elementary abelian group A acts on the group G where A and G have relatively prime orders. If CG(a) belongs to some formation F for all non-identity elements a in A, does it follow that G belongs to F? For many formations, the answer is shown to be yes provided that the rank of A is sufficiently large.

MSC classification

Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 1 , February 1983 , pp. 71 - 91
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Aschbacher, M. and Seitz, G., ‘On groups with a standard component known type’. Osaka Math. J. 13 (1976), 439–482.Google Scholar
[2]Berger, T., ‘Nilpotent fixed point free automorphism groups of solvable groups’, Math. Z. 131 (1973), 305–312.CrossRefGoogle Scholar
[3]Feit, W. and Thompson, J., ‘Solvability of groups of odd order’, Pacific J. Math. 13 (1963), 755–1029.Google Scholar
[4]Finkelstein, L., ‘Finite groups with a standard component of type J4’. Pacific J. Math. 71 (1977), 41–56.CrossRefGoogle Scholar
[5]Glauberman, G., ‘On solvable signalizer functors in finite groups’, Proc. London Math. Soc. (3) 33 (1976), 1–27.CrossRefGoogle Scholar
[6]Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
[7]Gross, F., ‘A note on fixed-point-free solvable operator groups’, Proc. Amer. Math. Soc. 19 (1968), 1363–1365.CrossRefGoogle Scholar
[8]Gross, F., ‘Elementary abelian operator groups’, Bull. Austral. Math. Soc. 7 (1972), 91–100.CrossRefGoogle Scholar
[9]Hall, P., ‘Theorems like Sylow's’, Proc. London Math. Soc. (3) 6 (1956), 286–304.CrossRefGoogle Scholar
[10]Huppert, B., Endliche Gruppen 1 (Springer, Berlin, 1967).CrossRefGoogle Scholar
[11]Jones, W., private communication.Google Scholar
[12]Schmid, P., ‘Every saturated formation is a local formation’, J. Algebra 51 (1978), 144–148.CrossRefGoogle Scholar
[13]Steinberg, R., Lectures on Chevalley groups, (Yale University, 1967).Google Scholar
[14]Ward, J., ‘On finite groups admitting automorphisms with nilpotent fixed-point group’, Bull. Austral. Math. Soc. 5 (1971), 281–282.CrossRefGoogle Scholar
[15]Ward, J., ‘On finite soluble groups and the fixed point groups of automorphisms’. Bull. Austral. Math. Soc. 5 (1971), 375–378.CrossRefGoogle Scholar
[16]Ward, J., ‘Nilpotent signalizer functors on finite groups’, Bull. Austral. Math. Soc. 9 (1973), 367–377.CrossRefGoogle Scholar