Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 145 / 4

Let $Z=Z(G)$. The inclusion ${\mathbb{F}}_p[[Z]]\subseteq Z({\mathbb{F}}_p[[G]])$ is immediate. For the reverse inclusion, project a central element to every finite group algebra ${\mathbb{F}}_p[G/U]$, where $U$ ranges over open normal subgroups. Its coefficients are constant on conjugacy classes. Compatibility as $U$ shrinks shows that a nonzero coefficient can persist only on an element with finite conjugacy class in $G$: an infinite conjugacy orbit splits into arbitrarily large p-power collections in finer quotients, whose fibre sums vanish in characteristic $p$.

In a p-valued group, an element with finite conjugacy class is central. Indeed, its centralizer is open, so some p-power of every element centralizes it; the p-valuation and the leading commutator identity then force the original commutators to vanish. Thus the finite-conjugacy center is $Z$, and the compatible finite-quotient expansions are supported on $Z$. This proves the Center of an Iwasawa algebra of a complete p-valued group:

For $G=1+p{M}_2({\mathbb{Z}}_p)$, a matrix commuting with every elementary principal-congruence matrix commutes with the full matrix algebra and is scalar. HenceConsequently