J. Aust. Math. Soc.  79 (2005), 213-229
On properties of group closures of one-to-one transformations

Inessa Levi
  Department of Mathematics
  Morgan Hall 118
  1 University Circle
  Western Illinois University
  Macomb, IL 61455-1390
  USA
  I-Levi@wiu.edu


Abstract
For a permutation group $H$ on an infinite set $X$ and a transformation $f$ of $X$, let $\langle f : H \rangle = \langle \{hfh^{-1}: h \in H \} \rangle$ be a group closure of $f$. We find necessary and sufficient conditions for distinct normal subgroups of the symmetric group on $X$ and a one-to-one transformation $f$ of $X$ to generate distinct group closures of $f$. Amongst these group closures we characterize those that are left simple, left cancellative, idempotent-free semigroups, whose congruence lattice forms a chain and whose congruences are preserved under automorphisms.
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