The kernel of the adjoint representation of a p-adic Lie group need not have an abelian open normal subgroup

Abstract

Let G be a p-adic Lie group and Ad be the adjoint representation of G on its Lie algebra. It was claimed in the literature that the kernel K of Ad always has an abelian open normal subgroup. We show by means of a counterexample that this assertion is false; it can even happen that K=G but G has no abelian subnormal subgroup except for the trivial group. The arguments are based on auxiliary results on subgroups of free products with central amalgamation.