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 \(\varvec{N}\) of order \(\varvec{p}\). The first author and Zalesskii have previously shown that a \(\mathbb {Z}_p\) \(\varvec{G}\)-lattice is a permutation module if, and only if, its \(\varvec{N}\)-invariants, its \(\varvec{N}\)-coinvariants, and a third module are all G/N permutation modules over \(\mathbb {Z}_p, \mathbb {Z}_p\) and \(\mathbb {Z}_p\) 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 \(\mathbb {Z}_p\) \(\varvec{G}\)-lattice for an abelian \(\varvec{p}\)-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 \(\varvec{N}\)-invariants and the \(\varvec{N}\)-coinvariants of a given lattice \(\varvec{U}\) are permutation modules, then so is \(\varvec{U}\). 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.