THE KERNEL OF THE ADJOINT REPRESENTATION OF A p-ADIC LIE GROUP NEED NOT HAVE AN ABELIAN OPEN NORMAL SUBGROUP

摘要

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.