Exponents of almost simple groups and an application to the restricted Burnside problem (Q1191440)

The restricted Burnside problem \(R(n)\) conjectures that for each \(r\) there are only finitely many \(r\)-generator, finite groups of exponent \(n\). Zelmanov has recently announced an affirmative answer to \(R(p^ e)\), \(p^ e\) a prime power. In the paper under review the authors prove a theorem that together with Zelmanov's result, a theorem of \textit{P. Hall} and \textit{G. Higman} [Proc. Lond. Math. Soc., III. Ser. 6, 1-42 (1956; Zbl 0073.255)] one gets the affirmatice answer to \(R(n)\) for all \(n\). The discussion of the paper rests on the following main theorem: Let \(G\) be a finite group such that \(E(G) \neq 1\), \(G/E(G)C_ G(E(G))\) is cyclic of order \(2^ a\) for some \(a\). Then there is a 2-element \(g\in G\) such that \(G = E(G)C_ G(E(G))\langle g\rangle\) and \(g^{2^ a} \not\in C_ G(E(G))\). This theorem is proved by reduction to the case that \(F^*(G)\) is simple and a detailed study of almost simple groups of linear, unitary or orthogonal type.