On properties of group closures of one-to-one transformations.

For a permutation group H on an infinite set X and a transformation / of X, let (/:// ) = ({hfh\u271 : h € //)) be a group closure of / . We find necessary and sufficient conditions for distinct normal subgroups of the symmetric group on X and a one-to-one transformation / of X to generate distinct group closures of / . 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