On completely reducible solvable subgroups of GL(n, Δ)

Let Δ be a finite field and denote by GL(n, Δ) the group of n×n nonsingular matrices defined over Δ. Let R⊆GL(n, Δ) be a solvable, completely reducible subgroup of maximal order. For |Δ|≧2, |Δ|≠3 we give bounds for |R| which improve previous ones. Moreover for |Δ|=3 or |Δ|>13 we determine the structure of R, in particular we show that R is unique, up to conjugacy.