Periodic modules of large periods for metacyclic \(p\)-groups (Q1178872)

Let \(G=M_{m(p)}\), \(m\geq 3\) be the non-abelian metacyclic \(p\)-group, \(p>2\) given by the presentation \[ M_{m(p)}=\langle x,y\mid x^{p^{m- 1}}=1=y^ p, y^{-1}xy=x^{1+p^{m-2}}\rangle, \] and let \(k\) be a field of characteristic \(p\). It is known from earlier work that a periodic \(kG\)-module must have period dividing \(2p\). Here the authors exhibit a submodule of the \(2p\)-th syzygy \(\Omega^{2p}(k)\) of the trivial module \(k\) which has period precisely \(2p\).