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.

Given a group \(G\), we write \(x^G\) for the conjugacy class of \(G\) containing the element \(x\). A famous theorem of B. H. Neumann states that if \(G\) is a group in which all conjugacy classes are finite with bounded size, then the derived group \(G'\) is finite. We establish the following result. Let \(n\) be a positive integer and \(K\) a subgroup of a group \(G\) such that \(|x^G|\leq n\) for each \(x\in K\). Let \(H=\langle K^G\rangle\) be the normal closure of \(K\). Then the order of the derived group \(H'\) is finite and \(n\)-bounded. Some corollaries of this result are also discussed.

Assume that \(G\) is a finite group. For every \(a, b \in\mathbb{N},\) we define a graph \(\Gamma_{a,b}(G)\) whose vertices correspond to the elements of \(G^a\cup G^b\) and in which two tuples \((x_1,\dots,x_a)\) and \((y_1,\dots,y_b)\) are adjacent if and only if \(\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.\) We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about \(G\) is encoded by these graphs.

An element \(g\) of a group \(G\) is said to be right Engel if for every \(x\in G\) there is a number \(n=n(g,x)\) such that \([g,{}_{n}x]=1\). We prove that if a profinite group \(G\) admits a coprime automorphism \(\varphi\) of prime order such that every fixed point of \(\varphi\) is a right Engel element, then \(G\) is locally nilpotent.