On the structure of virtually nilpotent compact p-adic analytic groups
AbstractLetGbe a compactp-adic analytic group. We recall the well-understood finite radical{\Delta^{+}}and FC-centre Δ, and introduce ap-adic analogue of Roseblade’s subgroup{\mathrm{nio}(G)}, the unique largest orbitally sound open normal subgroup ofG. Further, whenGis nilpotent-by-finite, we introduce the finite-by-(nilpotentp-valuable) radical{\mathbf{FN}_{p}(G)}, an open characteristic subgroup ofGcontained in{\mathrm{nio}(G)}. By relating the already well-known theory of isolators with Lazard’s notion ofp-saturations, we introduce the isolated lower central (resp. isolated derived) series of a nilpotent (resp. soluble)p-valuable group, and use this to study the conjugation action of{\mathrm{nio}(G)}on{\mathbf{FN}_{p}(G)}. We emerge with a structure theorem forG,1\leq\Delta^{+}\leq\Delta\leq\mathbf{FN}_{p}(G)\leq\mathrm{nio}(G)\leq G,in which the various quotients of this series of groups are well understood. This sheds light on the ideal structure of the Iwasawa algebras (i.e. the completed group ringskG) of such groups, and will be used in future work to study the prime ideals of these rings.