In this paper the following question is discussed. Suppose that \(C\) is a finite cyclic \(p\)-group (\(p\) prime). Let \(\varphi\) be a faithful irreducible complex character of \(C\). In case of \(p\geq 3\), what can be said about the structure of a finite \(p\)-group \(\mathcal G\) containing \(C\) for which the induced character \(\varphi^{\mathcal G}\) is also irreducible?
The main result proved in this interesting paper, is the following: Let \(\mathcal G\) be a finite \(p\)-group (\(p\) odd prime) containing a cyclic subgroup \(C\). Consider a faithful irreducible character \(\varphi\) of \(C\) and suppose that \(\varphi^{\mathcal G}\) is irreducible, whereas also \(|C|\geq|{\mathcal G}:C|^2p\) and \(C\) a normal subgroup of \(H<{\mathcal G}\) with \(|{\mathcal G}:H|=p\). Then \(\mathcal G\) is determined by means of generators and relations, as supplied explicitly in the paper. There are two isomorphism types for \(\mathcal G\).