Rationally Determined Group Modules

Abstract

Green's correspondence of group modules finds its simplest expression when a finite multiplicative group $G$ has a trivial intersection Sylow $p$-subgroup $P$, for some prime $p$. Then it is between all isomorphism classes of projective-free $\mathbf{R}G$-lattices $\mathbf{L}$ and all isomorphism classes of projective-free $\mathbf{R}N$-lattices $\mathbf{K}$, where $\mathbf{R}$ is a suitable valuation ring and $N$ is the normalizer of $P$ in $G$. In that case we show in Theorem 3.2 below that the $\mathbf{R}G$-lattice $\mathbf{L}$ is determined by its associated lattices over the residue field and field of fractions of $\mathbf{R}$ if and only if $\mathbf{K}$ has this same property. By Theorem 3.7 some important $\mathbf{R}G$-lattices $\mathbf{L}$ have this property of being “rationally determined.” So it would be worthwhile to see if the $\mathbf{R}N$-lattices with this property (and perhaps with other properties preserved by this Green correspondence) could be classified.