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Commutators in linear groups. (English) Zbl 0676.20024
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 $H\sb 1,...,H\sb k$, and for each finite subset S of G there are elements $g\sb i$ in G such that $g\sb i\sp{-1}Sg\sb i\le H\sb i$ for $i=1,...,k$; (**) G contains a subgroup $G\sb 0$ and pairwise commuting subgroups $H\sb 1,...,H\sb k$, and for each finite subset S of G there are $g\sb i,g\sb{\sigma}\in G\sb 0$, where $i=1,...,k$ and $\sigma \in S\sb k$ (the symmetric group of degree k), such that $g\sb i\sp{-1}sg\sb i\in H\sb i$ and $g\sb{\sigma}\sp{-1}g\sb i\sp{-1}sg\sb ig\sb{\sigma}=g\sb{\sigma i}\sp{- 1}sg\sb{\sigma i}$ for $i=1,...,k$, $\sigma \in S\sb k$, and all $s\in S$. (a) If a group G satisfies the condition (*) with $k=2$, then c(G)$\le 3$. (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 $G\sb 0=G$, c(G)$\le 2$. (Proposition 1). Under the condition (**) with $k=5$, $G'$ is perfect and $c(G)=c(G')\le 2$; moreover, every element of $G'$ is the product of a commutator in G and a commutator in $G\sb 0$. (Theorem 2). Some applications are made to “infinite-dimensional” automorphism groups, including infinite-dimensional linear groups over an associative ring with unity.
Reviewer: Yu.I.Merzlyakov

20G35Linear algebraic groups over adèles and other rings and schemes
20F12Commutator calculus (group theory)
Full Text: DOI
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