# Non-existence of non-trivial normal elements in the Iwasawa Algebra of Chevalley groups

## TL;DR

This paper proves that in the Iwasawa algebra of certain Chevalley groups over zp, the only normal elements are units, removing previous restrictions on the prime p and confirming a conjecture.

## Contribution

It provides a new proof of the non-existence of non-trivial normal elements in the Iwasawa algebra, eliminating the 'nice prime' condition and confirming a conjecture.

## Key findings

- All non-zero normal elements are units in the Iwasawa algebra.
- The proof does not require the prime p to be 'nice'.
- The result confirms the conjecture by Ardakov, Wei, and Zhang.

## Abstract

For a prime $p>2$, let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_p$, $G(1)$ be the first congruence kernel of $G$ and $\Omega_{G(1)}$ be the mod-$p$ Iwasawa algebra defined over the finite field $\mathbb{F}_p$. Ardakov, Wei, Zhang have shown that if $p$ is a "nice prime " ($p \geq 5$ and $p \nmid n+1$ if the Lie algebra of $G(1)$ is of type $A_n$), then every non-zero normal element in $\Omega_{G(1)}$ is a unit. Furthermore, they conjecture in their paper that their nice prime condition is superfluous. The main goal of this article is to provide an entirely new proof of Ardakov, Wei and Zhang's result using explicit presentation of Iwasawa algebra developed by the second author of this article and thus eliminating the nice prime condition, therefore proving their conjecture.