  We give a simple proof of a major index determinant formula in the symmetric
group discovered by Krattenthaler and first proved by Thibon using
noncommutative symmetric functions. We do so by proving a factorization of an
element in the group ring of the symmetric group. By applying similar methods
to the groups of signed permutations and colored permutations, we prove
determinant formulas in these groups as conjectured by Krattenthaler.
