Relative subgroups in Chevalley groups

Abstract

We finish the proof of the main structure theorems for a Chevalley group G (Φ, R ) of rank ≥ 2 over an arbitrary commutative ring R . Namely, we prove that for any admissible pair ( A, B ) in the sense of Abe, the corresponding relative elementary group E (Φ, R, A, B ) and the full congruence subgroup C (Φ, R, A, B ) are normal in G (Φ, R ) itself, and not just normalised by the elementary group E (Φ, R ) and that [ E (Φ, R ), C (Φ, R, A, B )] = E , (Φ, R, A, B ). For the case Φ = F 4 these results are new. The proof is new also for other cases, since we explicitly define C (Φ, R, A, B ) by congruences in the adjoint representation of G (Φ, R ) and give several equivalent characterisations of that group and use these characterisations in our proof.