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Permutation representations and rational irreducibility

Published online by Cambridge University Press:  17 April 2009

John D. Dixon
John D. Dixon
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa ON K1S 5B6, Canada
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The natural character π of a finite transitive permutation group G has the form 1G + θ where θ is a character which affords a rational representation of G. We call G a QI-group if this representation is irreducible over ℚ. Every 2-transitive group is a QI-group, but the latter class of groups is larger. It is shown that every QI-group is ¾-transitive and primitive, and that it is either almost simple or of affine type. QI-groups of affine type are completely determined relative to the 2-transitive affine groups, and partial information is obtained about the socles of simply transitive almost simple QI-groups. The only known simply transitive almost simple QI-groups are of degree 2k−1(2k − 1) with 2k − 1 prime and socle isomorphic to PSL(2, 2k).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 3 , June 2005 , pp. 493 - 503
Copyright
Copyright © Australian Mathematical Society 2005

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