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.
@article{AccShum21,title={A stronger form of Neumann’s BFC-theorem},author={Acciarri, Cristina and Shumyatsky, Pavel},journal={Israel Journal of Mathematics},volume={242},number={1},pages={269--278},doi={https://doi.org/10.1007/s11856-021-2133-1},year={2021},}
2019
PAMS19
Profinite groups with an automorphism whose fixed points are right Engel
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.
@article{AccKhukShum19,author={Acciarri, Cristina and Khukhro, Evgeny and Shumyatsky, Pavel},title={Profinite groups with an automorphism whose fixed points are right Engel},journal={Proceedings of the American Mathematical Society},volume={147},number={9},pages={3691--3703},year={2019},issn={0002-9939},doi={https://doi.org/10.1090/proc/14519},}
