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Hall Higman Type Theorems, I

Published online by Cambridge University Press:  20 November 2018

T. R. Berger
T. R. Berger*
Affiliation:
Trinity College, Hartford, Connecticut; University of Minnesota, Minneapolis, Minnesota
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Suppose Q is a q-group for a prime q and C ≦ Aut(Q) is cyclic of order pe for a prime p ≠ q. Let k be a splitting field for CQ, the semidirect product of C by Q, of characteristic r ≠ q. Let V be a faithful irreducible k[CQ]-module. The k[CQ]-module V has been widely studied. When r = p the situation outlined above is similar to the situation occurring in the proof of Theorem B of Hall and Higman [3]. When r ≠ p it is similar to the corresponding theorem of Shult [5]. In all but restricted cases V|C has a regular k[C]-direct summand.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 513 - 531
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Berger, T. R., Class two p-groups as fixed point free automorphism groups, Illinois. J. Math. 14 (1970), 121–149.Google Scholar
2. Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
3. Hall, P. and Higman, G., On the p-length of p-s oh able groups and reduction theorems for Burnside's problem, Proc. London Math. Soc. 6 (1956), 1–42.Google Scholar
4. Passman, D. S., Solvable half-transitive automorphism groups, J. Algebra 6 (1967), 285–304.Google Scholar
5. Shult, E., On groups admitting fixed point free abelian operator groups, Illinois J. Math. 9 (1965), 701–720.Google Scholar