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The Double Transitivity of a Class of Permutation Groups

Published online by Cambridge University Press:  20 November 2018

Ronald D. Bercov
Ronald D. Bercov*
Affiliation:
Cornell University and University of Alberta, Edmonton
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Certain finite groups H do not occur as a regular subgroup of a uniprimitive (primitive but not doubly transitive) group G. If such a group H occurs as a regular subgroup of a primitive group G, it follows that G is doubly transitive. Such groups H are called B-groups (8) since the first example was given by Burnside (1, p. 343), who showed that a cyclic p-group of order greater than p has this property (and is therefore a B-group in our terminology).

Burnside conjectured that all abelian groups are B-groups. A class of counterexamples to this conjecture due to W. A. Manning was given by Dorothy Manning in 1936 (3).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 480 - 493
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Burnside, W., Theory of groups of finite order, 2nd ed. (Cambridge, 1911).Google Scholar
2. Kochendörffer, R., Untersuchungen iiber eine Vermutung von W. Burnside, Schriften math. Sem. Inst, angew. Math. Univ. Berlin, 3 (1937), 155–180.Google Scholar
3. Manning, D., On simply transitive groups with transitive abelian subgroups of the same degree, Trans. Amer. Math. Soc, 40 (1936), 324–342.Google Scholar
4. Schur, I., Zur Théorie der einfach transitiven Permutationsgruppen, Sitzungsberichte Preuss. Akad. Wiss., phys.-math. Kl. (1933), 598-623.Google Scholar
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8. Wielandt, H., Permutation groups, Lectures at the University of Tübingen, 1954-55, prepared by J. Andre, translated from the German by R. Bercov (Cal. Tech., 1962).Google Scholar