Tensor products of irreducible projective integer-valued \(p\)-adic representations of a cyclic \(p\)-group (Q2730521)

Let \(G\) be a finite group, let \(\mathbb{Z}_p\) be the ring of \(p\)-adic integers, and let \(\Gamma\) be an irreducible projective \(\mathbb{Z}_p\)-representation of the group \(G\). Denote by \([\Gamma]\) the class of equivalent, over \(\mathbb{Z}_p\), irreducible projective \(\mathbb{Z}_p\)-representations of the group \(G\) and by \(B_1(G,\mathbb{Z}_p)\) the algebra generated by the set \(\{[\Delta_i]\}\), where \(\{\Delta_i\}\) is the set of all irreducible projective \(\mathbb{Z}_p\)-representations of the group \(G\). The author studies the algebra \(B_1(G,\mathbb{Z}_p)\), where \(G\) is a cyclic group of order \(p^n\) (\(p\neq 2\), \(n\geq 1\)), and proves that \(B_1(G,\mathbb{Z}_p)\) is finite-dimensional, semi-simple and \(\dim_\mathbb{Q} B_1(G,\mathbb{Z}_p)=\sum_{k=0}^n (2k+1)\varphi(p^{n-k})\).