Irreducibilities of the induced characters of cyclic \(p\)-groups. (English) Zbl 0989.20009
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 \(\mathcal G\).
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 \(\mathcal G\).
Reviewer: R.W.van der Waall (Amsterdam)