On the Lie theory of \(p\)-adic analytic groups
In his paper [Groupes analytiques \(p\)-adiques, Publ. Math., Inst. Hautes Étud. Sci. 26, 389-603 (1965; Zbl 0139.02302)], \textit{M.~Lazard} gave the concept of a \textit{saturable pro-p-group}. This paper was reinterpreted by Lubotzky and Mann, whose concepts can be found in [\textit{J.~Dixon, M.~du Sautoy, A.~Mann} and \textit{D.~Segal}, Analytic pro-\(p\) groups, 2nd edition, Cambridge University Press (1999; Zbl 0934.20001)]. They are starting with powerful pro-\(p\)-groups. It was an open question whether certain powerful pro-\(p\)-groups, namely the \textit{uniform} ones, are exactly the saturable pro-\(p\)-groups of Lazard. The author shows that this is not the case. He points out that Sylow-pro-\(p\) subgroups of many classical groups are saturable, but need not be uniformly powerful. Moreover, he deals with a correspondence between subgroups and Lie sublattices of saturable pro-\(p\)-groups established by \textit{I.~Ilani} [J. Algebra 176, No. 1, 34--58 (1995; Zbl 0834.20022)] by improving and extending Ilani's argument. This is done to prepare additional research in the subject of \(p\)-adic analytic groups.
saturable subgroups, uniformly powerful subgroups, Lie sublattices
scientific article; zbMATH DE number 2156071