Let V be a complex representation of a finite group G of order g. Derksen conjectured that the pth syzygies of the invariant ring Sym(V ) are generated in degrees ⤠(p+ 1)g. We point out that a simple application of the theory of twisted commutative algebras â using an idea due to Weyl â gives the weaker bound pg, almost for free. Fix a finite group G of order g. Let V be a finite dimensional complex representation of G and put R = Sym(V )G, which we regard as a graded ring. Let E â R be a homogeneous vector subspace generating R as an algebra, and put S = Sym(E), so that S â R is a surjection of graded algebras. The space Torp (R,C) is then a graded vector space, called the space of p-syzygies of R. We let sp(V ;E) be the maximal degree occurring in it. One can show that if E â EⲠthen sp(V ;E) ⤠sp(V ;Eâ˛). Furthermore, sp(V ;E) is independent of E if E is chosen to be minimal; we denote this common value by sp(V ). We then have the following conjecture of Derksen (see [D, Conj. 3] for a more precise version): Conjecture 1. We have sp(V ) ⤠(p+ 1)g for any V . Derksen [D, Thm. 2] proved the conjecture for p = 1, but the general case is open. To state our main result, we first introduce some notation. Let β(V ) be the minimal integer such that R is generated in degrees ⤠β(V ). Let β be the maximum value of β(V ) over all V , the so-called Noether number of G. Noetherâs theorem [W, §3] states that β ⤠g. Let d1, . . . , dn be the degrees of the irreducible representations of G and let m be the sum of the di. Finally, put δp = (β â 1)g â (mâ 1)βp; note that this is negative for p 0. We then have: Theorem 2. We have sp(V ) ⤠β2mp+ δp for any V . Although this is weaker than the conjecture, the bound is significant since it is independent of V . Using the fact that β and m are both bounded by g, we deduce the following corollary: Corollary 3. We have sp(V ) ⤠pg3 for any V . Remark 4. In fact, m and β are often strictly smaller than g, sometimes significantly so. We have m ⤠âng, where n is the number of conjugacy classes in G. Typically, n is much smaller than g; for example, if G is a symmetric group then n = O(g ), for any > 0. The currently known bounds on β are not very sharp. A result of CziszterâDomokos [CD] states that β < 1 2g unless G has a cyclic subgroup of index two, or is one of four exceptions. A conjecture of Pawale [W, Conj. 3.9] asserts that if G is the semi-direct product of cyclic groups of prime orders p and q, with q | pâ 1, then β = p+ q â 1; note that m = p+ q â 1 in this case as well. When p and q are approximately equal, this gives m = β = O( â g). Thus, in many case, our theorem gives a bound of the form sp(V ) = O(pg θ) with θ < 3. Remark 5. Theorem 2 (and our proof of it) is valid over any field of characteristic zero. Derksen also proposed his conjecture in positive characteristic not dividing g. It would be interesting if our proof could be adapted to work in this setting. We now prove the theorem. Put sp(V ) = sp(V ;E) where E = âβ i=1Ri. As discussed, we have sp(V ) ⤠sp(V ), so it suffices to bound the latter. The key result is the following lemma. Date: November 2, 2012. The author was supported by NSF fellowship DMS-0902661. 1