Abstract:
We obtain some sufficient conditions for potency and virtual potency for automorphism groups and the split extensions of some groups. In particular, considering a finitely generated group $G$ residually $p$-finite for every prime $p$, we prove that each split extension of $G$ by a torsion-free potent group is a potent group, and if the abelianization rank of $G$ is at most $2$ then the automorphism group of $G$ is virtually potent. As a corollary, we derive the necessary and sufficient conditions of virtual potency for certain generalized free products and HNN-extensions.
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