Groups in which squares have boundedly many conjugates
The authors are interested in the implications of bounding the sizes of certain conjugacy classes in a group. An early result in this direction is due to \textit{B. H. Neumann} [J. Lond. Math. Soc. 29, 236--248 (1954; Zbl 0055.01604)] and says that if all conjugacy class sizes in a group \(G\) are finite and of bounded size then the derived group \(G'\) is finite. In particular, denoting the conjugacy class of \(x \in G\) by \(x^G\), if \(|x^G| \leq n\) for all \(x \in G\) then \(|G'|\) is \(n\)-bounded. The authors have previously considered the case when conjugacy class sizes of commutators are bounded by \(n\) and concluded that in this case \(G''\) is finite and its order is \(n\)-bounded. In the present paper, the authors continue this investigation, in particular they consider the implications of bounding the conjugacy class sizes of square elements. An element \(x \in G\) is called a square if there exists \(y \in G\) such that \(x = y^2\). The main result of the paper under review is the following. Let \(H\) denote the subgroup generated by all square elements and suppose \(|x^H| \leq n\) for any square \(x \in G\). Then \(\gamma_3(H)\), the third term of the lower central series of \(H\), is finite and \(n\)-bounded. The authors leave as an open question whether \(H'\) is finite and \(n\)-bounded if \(|x^G| \leq n\) for any square \(x \in G\).
scientific article; zbMATH DE number 6998942
scientific article