Fusion in profinite groups
The first aim of the paper under review is to establish the basic theory of fusion and transfer for profinite groups, extending and generalizing the classical theory for finite groups. (For the latter, see for instance Sections~2 and 3 of the book by \textit{D. Gorenstein, R. Lyons} and \textit{R. Solomon} [The classification of the finite simple groups. No. 2. Part I, Chapter G: General group theory, (Mathematical Surveys and Monographs. Vol. 40, No. 2, Am. Math. Soc., Providence, RI) (1996; Zbl 0836.20011)].) The following question is then considered. Suppose \(\mathcal P\) is a class of pro-\(p\) groups, for some prime \(p\). What properties of \(\mathcal P\) imply that if the group \(S\in\mathcal P\) is isomorphic to a Sylow \(p\)-subgroup (or a weakly/strongly closed subgroup thereof) of the profinite group \(G\), then it controls transfer or fusion in \(G\)? An interesting result in this direction is the following. If the Sylow \(p\)-subgroup \(S\) of the profinite group \(G\) is powerful, then \(S\) controls transfer in \(G\). For the theory of powerful (pro-)\(p\)-groups, see [\textit{J. D. Dixon, M. P. F. du Sautoy, A. Mann} and \textit{D. Segal}, Analytic pro-\(p\) groups. 2nd ed., (Cambridge Studies in Advanced Mathematics 61, Cambridge University Press, Cambridge) (1999; Zbl 0934.20001)].
scientific article; zbMATH DE number 1472058