arXiv:2611.13133v1 Announce Type: new Abstract: It was shown in a recent work by Senise and Shumyatsky that if $G$ is a finite group such that $|x^G|\leq m$ whenever $x$ is a commutator of prime power order, then $G''$ has $m$-bounded order. In this paper, we examine the following question: given a profinite group $G$ such that $|x^G|$ is finite whenever $x$ is a commutator of prime power order, does it follow that $G''$ is finite? We show that, in general, the answer to this is negative. The main result obtained in this paper is the following theorem: Let $G$ be a profinite group in which $|x^G|$ is finite whenever $x$ is a commutator of prime power order. Assume that $\pi(G)$ is finite. Then: (1) The second commutator subgroup $G''$ is finite; (2) $G$ has an open normal subgroup of nilpotency class at most $3$.
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