Extension and the irreducibilities of the induced characters of cyclic \(p\)-groups. (Q1860642)

A survey is given around the following two problems: I. Let \(p\) be an odd prime, \(\varphi\in\text{Irr}(C)\) and \(\varphi\) faithful, where \(C\) is a cyclic subgroup of order \(n\) of the \(p\)-group \(P\). Suppose the induced character \(\varphi^P\) is an irreducible character of \(P\). What does \(P\) look like? II. The same question as in I, but now for \(p=2\), and \(C\) replaced by either a generalized quaternion group, a dihedral group, or a semi-dihedral group. In this paper, Problem I is treated. The purpose of this paper is to determine the groups \(N_P(C)\), \(N_P(N_P(C))\) and \(N_P(N_P(N_P(C)))\) in the case where \(|N_P(C):C|=p\) and \(|P:C|\geq p^3\).