Irreducibility of an affine space in algebraic geometry over a group

We prove a theorem which states that if $G$ is an equationally Noetherian group that is locally approximated by finite $p$-groups for each prime $p$ then an affine space $G^n$ in a respective Zariski topology is irreducible for any $n$. The hypothesis of the theorem is satisfied by free groups, free soluble groups, free nilpotent groups, finitely generated torsion-free nilpotent groups, and rigid soluble groups. Also we introduce corrections to a lemma on valuations, which has been used in some of the author's previous works.