Cristina Acciarri

A group-word \(w\) is concise in a class of groups \(\mathcal X\) if and only if the verbal subgroup \(w(G)\) is finite whenever \(w\) takes only finitely many values in a group \(G\in \mathcal X\). It is a long-standing open problem whether every word is concise in residually finite groups. In this paper we observe that the conciseness of a word \(w\) in residually finite groups is equivalent to that in the class of virtually pro-\(p\) groups. This is used to show that if \(q,n\) are positive integers and \(w\) is a multilinear commutator word, then the words \(w^q\) and \([w^q,_{n} y]\) are concise in residually finite groups. Earlier this was known only in the case where \(q\) is a prime power. In the course of the proof we establish that certain classes of groups satisfying the law \(w^q\equiv1\), or \([\delta_k^q,{}_n\, y]\equiv1\), are varieties.

Let \(G\) be a finite group admitting a coprime automorphism \(\alpha\) of order \(e\). Denote by \(I_G(\alpha)\) the set of commutators \(g^{-1}g^\alpha\), where \(g\in G\), and by \([G,\alpha]\) the subgroup generated by \(I_G(\alpha)\). We study the impact of \(I_G(\alpha)\) on the structure of \([G,\alpha]\). Suppose that each subgroup generated by a subset of \(I_G(\alpha)\) can be generated by at most \(r\) elements. We show that the rank of \([G,\alpha]\) is \((e,r)\)-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of \(I_G(\alpha)\) has odd order, then \([G,\alpha]\) has odd order too. Further, if every pair of elements from \(I_G(\alpha)\) generates a soluble, or nilpotent, subgroup, then \([G,\alpha]\) is soluble, or respectively nilpotent.