On profinite groups in which commutators are covered by finitely many subgroups

Abstract

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all ${\gamma }_k$-values in $G$, it is shown that ${\gamma }_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contains all ${\gamma }_k$-values, it is shown that ${\gamma }_k(G)$ has finite $(k,r,s)$-bounded rank.

13 pages, submitted

Cited by in corpus (1)

• On verbal subgroups in finite and profinite groups