Nonsplit extensions of indecomposable modules of \(p\)-integral representations of cyclic groups of order \(p^2\) (Q2759403)

Let \(R\) be a principal ideal domain which is a subring of a field \(F\); let \(G\) be a finite group; let \(\Gamma\) be an explicit \(R\)-representation of the group \(G\); let \(M\) be the \(RG\)-module of the representation \(\Gamma\); let \(FM\) be a linear space over the field \(F\) such that \(M\subset FM\) and let every \(R\)-basis in \(M\) be an \(F\)-basis in \(FM\). Let \(\widehat M=FM^+/M^+\) be the factor-group of the additive group \(FM^+\) with respect to the subgroup \(M^+\), and let \(f\) be a \(1\)-cocycle of the group \(G\) with values in the group \(\widehat M\). By \(K(G,M,f)\) we denote the group with elements \((g,x)\), where \(g\in G\), \(x\in f(g)\), \(f(g)\in\widehat M\) and with operation \((g,x)(g',y)=(gg',gy+x)\), where \(g,g'\in G\), \(x\in f(g)\), \(y\in f(g')\). In this article all nonsplit extensions of \(K(G,M,f)\) are described in the case \(G=H=\langle a\rangle\), the cyclic group of order \(p^2\); \(R=I_p\) the ring of \(p\)-integral rational numbers (or the ring of \(p\)-adic numbers); \(M\) the module of an indecomposable \(I_p\)-representation of the group \(H\).