Commutators in linear groups
Let G be a group, and c(G) denote the least natural number c such that every element of the commutator subgroup \(G'\) of G is a product of c commutators. The following conditions on G are considered in the article: (*) G contains pairwise commuting subgroups \(H_ 1,...,H_ k\), and for each finite subset S of G there are elements \(g_ i\) in G such that \(g_ i^{-1}Sg_ i\leq H_ i\) for \(i=1,...,k\); (**) G contains a subgroup \(G_ 0\) and pairwise commuting subgroups \(H_ 1,...,H_ k\), and for each finite subset S of G there are \(g_ i,g_{\sigma}\in G_ 0\), where \(i=1,...,k\) and \(\sigma \in S_ k\) (the symmetric group of degree k), such that \(g_ i^{-1}sg_ i\in H_ i\) and \(g_{\sigma}^{-1}g_ i^{-1}sg_ ig_{\sigma}=g_{\sigma i}^{- 1}sg_{\sigma i}\) for \(i=1,...,k\), \(\sigma \in S_ k\), and all \(s\in S\). (a) If a group G satisfies the condition (*) with \(k=2\), then c(G)\(\leq 3\). (b) If a group G satisfies (*) with \(k=3\), then every commutator in G is a commutator in \(G'\). So \(G'\) is perfect and \(c(G)=c(G')\). (c) Under the condition (**) with \(k=3\) and \(G_ 0=G\), c(G)\(\leq 2\). (Proposition 1). Under the condition (**) with \(k=5\), \(G'\) is perfect and \(c(G)=c(G')\leq 2\); moreover, every element of \(G'\) is the product of a commutator in G and a commutator in \(G_ 0\). (Theorem 2). Some applications are made to ``infinite-dimensional'' automorphism groups, including infinite-dimensional linear groups over an associative ring with unity.
scientific article; zbMATH DE number 4107091
scientific article