[For the entire collection see Zbl 0731.00008.]
For linear representations of a finite group \(G\) on a finite dimensional vector space \(V\) over an algebraically closed field of characteristic zero consider the ring \({\mathcal O}(V)^ G\) of \(G\)-invariant polynomial functions. By Noether’s theorem, \({\mathcal O}(V)^ G\) is generated by the homogeneous invariants of degree smaller or equal to the order of the group \(G\). For cyclic groups \(G\), any system of homogeneous generators must include invariants of degree at least equal to the order of \(G\). However, for non-cyclic groups \({\mathcal O}(V)^ G\) can be generated by homogeneous functions of degree strictly less than the order of \(G\). By \(\beta(V,G)\) we denote the smallest positive integer \(n\) such that the homogeneous invariants of degree \(\leq n\) generate the ring \({\mathcal O}(V)^ G\). For the regular representation \(V_{\text{reg}}\) the inequality \(\beta(V,G)\leq\beta(V_{\text{reg}},G)\) holds for any representation \(V\) of \(G\). We show \(\beta(V_{\text{reg}},G)<\) order of \(G\) for a non-cyclic finite group \(G\). For the dihedral groups, the quaternion group, the alternating group \(A_ 4\) and some abelian groups we give more explicit results.