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

TL;DR

This paper provides a counterexample demonstrating that the kernel of the adjoint representation of a p-adic Lie group need not contain an abelian open normal subgroup, challenging previous assumptions in the literature.

Contribution

It shows that the previously claimed universal property of the kernel of the adjoint representation does not hold, using counterexamples involving free products with central amalgamation.

Findings

Counterexample where kernel equals the entire group but lacks nontrivial abelian subnormal subgroups

Disproof of the claim that the kernel always has an abelian open normal subgroup

Insights into subgroup structures of free products with central amalgamation

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.