Abstract
Let G be a p-adic Lie group with Lie algebra 𝔤 and Ad: G → Aut(𝔤) be the adjoint representation. It was claimed in the literature that the kernel K≔ker(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.
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