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On Simply Transitive Primitive Permutation Groups

Published online by Cambridge University Press:  20 November 2018

R. D. Bercov
R. D. Bercov*
Affiliation:
University of Alberta, Edmonton, Alberta
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In (1) we considered finite primitive permutation groups G with regular abelian subgroups H satisfying the following hypothesis:

(*) H = A × B × C, where A is cyclic of prime power order pα ≠ 4, B has exponent pβ < pα, and C has order prime to p.

We remark that an abelian group fails to satisfy (*) (apart from the minor exception associated with the prime 2) if and only if it is the direct product of two subgroups of the same exponent.

We showed in (1) that such a group G is doubly transitive unless it is the direct product of two or more subgroups each of the same order greater than 2. This was done by showing that (in the terminology of (3)) the existence of a non-trivial primitive Schur ring over H implies such a direct decomposition of H.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1062 - 1068
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bercov, R. D., The double transitivity of a class of permutation groups, Can. J. Math. 17 (1965), 480–493.Google Scholar
2. Schur, I., Zur théorie der einfach transitiviten permutationsgruppen, Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. 1933, 598–623.Google Scholar
3. Wielandt, H., Finite permutation groups (Academic Press, New York, 1964).Google Scholar