On the irreducible modular representations of finite \(p\)-groups over commutative local rings (Q2730578)

This article deals with the irreducible matrix representations of finite \(p\)-groups over a commutative Noetherian local ring \(K\) of characteristic \(p^s\) (\(s>0\)), which contains a nonzero nilpotent element. Let \(G\) be a finite \(p\)-group of order \(|G|>1\). Denote by \(\text{Rad }K\) the Jacobson radical of the ring \(K\). The authors prove the following theorem. The set of the degrees of all irreducible matrix \(K\)-representations of the group \(G\) is finite if and only if one of the following conditions is fulfilled: (1) \(|G|=p\), \(s>1\), \(\text{Rad }K=pK\), the set of the degrees of all irreducible polynomials over the field \(K/\text{Rad }K\) is finite; (2) \(|G|=2\), \(s=1\), \((\text{Rad }K)^2=0\), \(K\) is a principal ideal ring, the set of the degrees of all irreducible polynomials over the field \(K/\text{Rad }K\) is finite.