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Nilpotent Partition-Inducing Automorphism Groups

Published online by Cambridge University Press:  20 November 2018

Martin R. Pettet
Martin R. Pettet*
Affiliation:
Texan A&M University, College Station, Texas
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If A is a group acting on a set X and x ∈ X, we denote the stabilizer of x in A by CA(x) and let Γ(x) be the set of elements of X fixed by CA(x). We shall say the action of A is partitive if the distinct subsets Γ(x), x ∈ X, partition X. A special example of this phenomenon is the case of a semiregular action (when CA (x) = 1 for every x ∈ X so the induced partition is a trivial one). Our concern here is with the case that A is a group of automorphisms of a finite group G and X = G#, the set of non-identity elements of G. We shall prove that if A is nilpotent, then except in a very restricted situation, partitivity implies semiregularity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 412 - 420
Copyright
Copyright © Canadian Mathematical Society 1981

References

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