The automorphism groups of relatively free algebras
The authors study the automorphisms groups of relatively free algebras having in mind the automorphism groups of relatively free groups. They saw that the results can be formulated by considering relatively free algebras of infinite rank. So let \(\Omega\) be an operator domain. An \(\Omega\)-algebra is a set \(G\) in which for every \(\omega\in\Omega\) there is an \(n\)-ary operation of \(G\) for some integer \(n\). If \(X\) is a subset of \(G\), then \(G\) is called relatively free on \(X\), and \(X\) a free generating set for \(G\), if \(X\) generates \(G\) and every function \(X\to G\) can be extended to an \(\Omega\)-algebra homomorphism \(G\to G\). If \(U\) is a subset of \(\Gamma=\Aut(G)\), then \(\Gamma_{(U)}\) denotes the subgroup of \(\Gamma\) consisting of the pointwise stabilizers of \(U\) and \(\Gamma\) becomes a topological group where the open sets are the unions of cosets of subgroups of this form with \(U\) finite. The authors prove the existence of large dense free subgroups of \(\Gamma\) for \(G\) of infinite rank \(| X|=\kappa\). If \(G\) is an \(\Omega\)-algebra, then \(G\) is said to have the small index property if every subgroup of \(\Aut(G)\) with index less than \(2^{\aleph_0}\) is open. We can quote, with no further terminology, two of the main theorems of this paper: Let \(\Omega\) be an operator domain and let \(G\) be a relatively free \(\Omega\)-algebra of infinite rank \(\kappa\). Suppose that \(| G|\leq 2^\kappa\) and let \(\eta\) be any cardinal such that \(| G|\leq\eta\leq 2^\kappa\). Then \(\Aut(G)\) has a dense subset of cardinality \(\eta\) which freely generates a free subgroup of \(\Aut(G)\). Let \(\Omega\) be an operator domain and let \(G\) be a relatively free \(\Omega\)-algebra of infinite rank. If \(G\) has the small index property then \(\Aut(G)\) has no proper normal subgroup of index less than \(2^{\aleph_0}\). Hence \(\Aut(G)\) is perfect.
scientific article; zbMATH DE number 1247624
scientific article