Automorphisms of finite groups of bounded rank
The author proves that, if \(G\) is a finite group of rank \(r\) admitting an automorphism \(\alpha\) with \(m\) fixed points, then \(G\) has a soluble characteristic subgroup \(H\) whose index is bounded by a function of \(m\) and \(r\). Moreover, if the orders of \(G\) and \(\alpha\) are coprime, then the derived length of \(H\) is also bounded by a function of \(m\) and \(r\). In order to show that the derived length of \(H\) is bounded, the author reduces to a Lie-theoretic problem, and this reduction is based on the construction of a particular Lie ring from a certain type of \(p\)-group. Concerning Lie rings, he proves that if \(L\) is a Lie ring of finite rank \(r\) admitting a semisimple fixed point-free automorphism \(\alpha\), then \(L\) is soluble with derived length at most \(2^{r-1} - 1\).
scientific article; zbMATH DE number 446719
scientific article