Group closures of partial transformations (Q1849730)

\(X\) is an infinite set and \({\mathcal G}_X\) denotes the group of all permutations of \(X\). For any transformation \(f\) of \(X\) and any subgroup \(H\) of \({\mathcal G}_X\), let \(\langle f:H\rangle\) be the semigroup of transformations of \(X\) which is generated by \(\{hfh^{-1}:h\in H\}\) and let \(G_{\langle f:H\rangle}=\{g\in{\mathcal G}_X:g\langle f:H\rangle g^{-1}=\langle f:H\rangle\}\). For a partial one-to-one transformation \(f\) of \(X\), let \(C_{{\mathcal G}_X}(f)=\{g\in{\mathcal G}_X:gf=fg\}\). The authors prove, among other things, that if \(f\) is a partial one-to-one transformation of \(X\) which is not a permutation and \(H\) is a normal subgroup of \({\mathcal G}_X\), then \(G_{\langle f:H\rangle}=H\) if and only if \(C_{{\mathcal G}_X}(f)\subseteq H\).