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Group closures of one-to-one transformations

Published online by Cambridge University Press:  17 April 2009

Inessa Levi
Inessa Levi
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY 40292, United States of America
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For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 177 - 188
Copyright
Copyright © Australian Mathematical Society 2001

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