Lower degree bounds for modular invariants and a question of I. Hughes

We prove two statements. The first one is a conjecture of Ian Hughes which states that iff ₁, ..., f_n are primary invariants of a finite linear groupG, then the least common multiple of the degrees of thef _i is a multiple of the exponent ofG.The second statement is about vector invariants: IfG is a permutation group andK a field of positive characteristicp such thatp divides |G|, then the invariant ringKV ^m]^G ofm copies of the permutation moduleV overK requires a generator of degreem(p–1). This improves a bound given by Richman 6], and implies that there exists no degree bound for the invariants ofG that is independent of the representation.