Group closures of one-to-one transformations (Q2758182)

Let \(X\) be an infinite set, let \({\mathcal T}_X\) denote the full transformation semigroup on \(X\) and let \({\mathcal G}_X\) denote the group of bijections on \(X\). Let \(H\) be a subgroup of \({\mathcal G}_X\) and let \(f\in{\mathcal T}_X\). The \(H\)-closure of \(f\) is denoted by \(\langle f:H\rangle\) and is defined by \(\langle f:H\rangle=\langle\{hfh^{-1}:h\in H\}\rangle\). It is the smallest subsemigroup of \({\mathcal T}_X\) that contains \(f\) and whose automorphism group contains all the inner automorphisms induced by elements of \(H\). For a subsemigroup \(S\) of \({\mathcal T}_X\), denote by \(G_S\) the subgroup of \({\mathcal G}_X\) consisting of all those permutations which preserve \(S\) under conjugation. The subsemigroup \(S\) is said to be \(H\)-normal where \(H\) is any subgroup of \({\mathcal G}_X\) if \(G_S=H\).NEWLINENEWLINENEWLINEThe author considers two problems in this paper. The first is to characterize the normal subgroups \(H\) of \({\mathcal G}_X\) and those \(f\in{\mathcal T}_X\) such that \(\langle f: H\rangle\) is \(H\)-normal and the second is to characterize those subsemigroups of \({\mathcal T}_X\) which have the inner automorphism property. She solves the first problem in the case where \(f\) is injective. The statement of the theorem is a bit too complicated to restate here. As for the second problem, she shows that if \(S\) is a semigroup of injections on \(X\), not all of which are bijections, and if the alternating group is a subgroup of \(G_S\), then each automorphism of \(S\) is inner and \(\Aut(S)\cong G_S\).