Exercise 1 (Maximal compact subgroups of G). A lattice in Qp is a finitelygenerated Zp-submodule of Qp that generates Qp as vector space. In particular, it’s free of rank n. Note that G acts transitively on the set of lattices in Qp . (i) Show that K = StabG(Zp ). (ii) Suppose that K ′ is a compact subgroup of G. Show that K ′ stabilises a lattice. (Hint: show that the K ′-orbit of Zp is finite...

Abstract In [16], Peterzil and Steinhorn proved that if a group G definable in an o -minimal structure is not definably compact, then contains torsion-free subgroup of dimension 1. We prove here p -adic analogue the Peterzil–Steinhorn theorem, special case abelian groups. Let be -adically closed field M . If compact there H 1 which compact. future paper we will generalize this to non-abelian