Extensions of cyclic p-groups which preserve the irreducibilities of induced characters
For a prime p, we denote by Bn the cyclic group of order pn. Let ϕ be a faithful irreducible character of Bn, where p is an odd prime. We study the p-group G containing Bn such that the induced character ϕG is also irreducible. Set [NG(Bn):Bn] = pm and [G:Bn] = pM. The purpose of this paper is to determine the structure of G under the hypothesis [NG(Bn):Bn]2d ≤ pn, where d is the smallest integer not less than M/m.