On Baer-Suzuki \(\pi\)-theorems.
All groups considered in this paper are finite. The Baer-Suzuki theorem for a group \(G\) states that if \(p\), \(x\) are a prime number and an element of \(G\) such that \(\langle x,x^g\rangle\) is a \(p\)-group for all \(g\in G\) then \(x\in O_p(G)\). -- In general we cannot replace \(p\) in the Baer-Suzuki theorem by an arbitrary set \(\pi\) of prime numbers. Let \(\pi\) be a set of prime numbers and \(m\) a nonnegative integer The author introduces the class \(\mathcal{BS}_\pi\) of groups \(G\) for which the Baer-Suzuki theorems holds for the set \(\pi\). Also he says: that Baer-Suzuki \((\pi,m)\) holds for a group \(G\) and writes \(G\in\mathcal{BS}_\pi^m\) whenever \(O_\pi(G)\) contains every element \(x\in G\) such that any \(m\) elements conjugate to \(x\) generate a \(\pi\)-group. In particular \(\mathcal{BS}_\pi=\mathcal{BS}_\pi^2\). The author finds a sufficient condition for the Baer-Suzuki \(\pi\)-theorem in terms of nonabelian composition factors. Also he obtains, resting on several results which base on the classification of simple groups, that the Baer-Suzuki \(\pi\)-theorem holds for every group if \(2\notin\pi\). He states the conjecture that the class of \(\mathcal D_\pi\)-groups is contained in the class \(\mathcal{BS}_\pi\) and proves that \(\mathcal D_\pi\)-groups are contained in the class \(\mathcal{BS}_\pi^4\).
scientific article; zbMATH DE number 5912841
scientific article