Butler’s Method Applied to \(\mathbb {Z}_p[C_p\times C_p]\)-Permutation Modules

Let G be a finite p-group with normal subgroup \(\varvec{N}\) $\mathit{N}$ of order \(\varvec{p}\) $\mathit{p}$ . The first author and Zalesskii have previously shown that a \(\mathbb {Z}_p\) ${\mathbb{Z}}_p$ \(\varvec{G}\) $\mathit{G}$ -lattice is a permutation module if, and only if, its \(\varvec{N}\) $\mathit{N}$ -invariants, its \(\varvec{N}\) $\mathit{N}$ -coinvariants, and a third module are all G/N permutation modules over \(\mathbb {Z}_p, \mathbb {Z}_p\) ${\mathbb{Z}}_p,{\mathbb{Z}}_p$ and \(\mathbb {Z}_p\) ${\mathbb{Z}}_p$ respectively. The necessity of the first two conditions is easily shown but the necessity of the third was not known. We apply a correspondence due to Butler, which associates to a \(\mathbb {Z}_p\) ${\mathbb{Z}}_p$ \(\varvec{G}\) $\mathit{G}$ -lattice for an abelian \(\varvec{p}\) $\mathit{p}$ -group a set of simple combinatorial data, to demonstrate the necessity of the conditions, using the correspondence to construct highly non-trivial counterexamples to the claim that if both the \(\varvec{N}\) $\mathit{N}$ -invariants and the \(\varvec{N}\) $\mathit{N}$ -coinvariants of a given lattice \(\varvec{U}\) $\mathit{U}$ are permutation modules, then so is \(\varvec{U}\) $\mathit{U}$ . Our approach, which is new, is to translate the desired properties to the combinatorial side, find the counterexample there, and translate it back to a lattice.