Groups associated with finite transformation semigroups
Let \(S\) be a semigroup of mappings (perhaps partial) on \(N\), a finite \(n\)-element set, and let \(G_S\) be the group of all permutations \(h\) on \(N\) such that \(hah^{-1}\in S\) for all \(a\in S\). It is known that if \(S\) consists of total mappings and \(A_n\subseteq G_S\) then \(G_S=S_n\), the full symmetric group. This is also true under other circumstances such as \(n\) not being divisible by 4. However, the main result of this paper shows that if \(n=2m\), \(m\) even, and \(u\) is a so-called \(x\)-nilpotent of rank \(m\) then the semigroup generated by the conjugates of \(u\) with respect to the alternating group \(A_n\) is strictly contained in that generated by arbitrary conjugates of \(u\). They go on to determine the semigroups \(S\) of partial mappings such that \(G_S=A_n\), find generating sets for such \(S\), and all the ideals, automorphisms and congruences on \(S\). Finally, they address and partly answer the question as to when the semigroup generated by a map \(a\) and its conjugates with respect to a group \(G\) is equal to the semigroup generated by \(G\cup\{a\}\) with \(G\) deleted.
scientific article; zbMATH DE number 1545256