A note on the degree bounds of the invariant ring

Let $G=C_p\times H$ be a finite group, where $C_p$ is a cyclic group of prime order $p$ and $H$ is a $p^{\mathrm{\prime }}$-group. Let $\mathbb{F}$ be an algebraically closed field in characteristic $p$. Let $V$ be a direct sum of $m$ non-trivial indecomposable $G$-modules such that the norm polynomials of the simple $H$-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring $\mathbb{F}[V]$ with actions of $G$. In this paper, by the periodicity property, we showed that $\mathbb{F}[V{]}^G$ is generated by $m$ norm polynomials together with homogeneous invariants of degree at most $m|G|-\mathrm{d}\mathrm{i}\mathrm{m}(V)$ and transfer invariants, which yields the well-known degree bound $\mathrm{d}\mathrm{i}\mathrm{m}$ $(V)\cdot (|G|-1)$. More precisely, we found that this bound gets less sharp as the dimensions of simple $H$-modules increase.