Let \(G\) be a finite group and let \(\varphi\) be one of its automorphisms. Suppose that \(g \cdot \varphi(g)^3 \cdot \varphi^2(g)^6 \cdot \varphi^3(g)^5 = 1,\) for every \(g \in G\). What can we say about the structure of the group \(G\)? It turns out that we can say quite a lot: \(G\) is abelian, unless it has an element of order \(3,5,13,239\) or \(4733\). In order to explain why this is the case, we first define identities of automorphisms. We then show that some of these identities naturally correspond with classical results in the literature: the restricted Burnside problem, the Frobenius conjecture, and the classification of \(n\)-abelian groups. We conclude by formulating some general theorems that were recently obtained by the speaker in joint work with Evgeny Khukhro.
Finite groups with an automorphism satisfying a given identity
22.03.2022 13:15 - 14:45
Organiser:
H. Grobner, A. Minguez-Espallargas, A. Mellit
Location: