Abstract:
Let $\pi$ be a set of primes. A group $G$ is said to be $\pi$-potent if, given an element $a \in G$ and a positive integer $\pi$-number $n$ dividing the order of $a$, there exists a homomorphism from $G$ onto a finite group that maps $a$ to an element of order $n$. It is proved that if $G$ is a finitely generated metabelian group (or a residually finite abelian group, or a residually finite nilpotent group of finite rank, or a residually finite metabelian FATR group), then the following statements are equivalent for $G$: (1) $G$ is $\pi$-potent; (2) $G$ has no $p$-radicable elements of infinite order for any $p \in \pi$; (3) $G$ is virtually $\pi$-potent. It is also proved that if $G$ is a residually finite soluble FATR group (or a finitely generated abelian-by-polycyclic group), then conditions (2) and (3) are equivalent for $G$.
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