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Extensions of cyclic \(p\)-groups which preserve the irreducibilities of induced characters. (English) Zbl 1253.20006

For a prime \(p\), denote by \(B_n\) the cyclic group of order \(p^n\). Let \(\varphi\) be a faithful irreducible character of \(B_n\), where \(p\) is an odd prime. In this paper \(p\)-groups \(G\) are studied containing \(B_n\) such that the induced character \(\varphi^G\) is also irreducible. Set \([N_G(B_n):B_n]=p^m\) and \([G:B_n]=p^M\). The purpose of this paper is to determine the structure of \(G\) under the hypothesis \([N_G(B_n):B_n]^{2d}\leq p^n\), where \(d\) is the smallest integer not less than \(M/m\). The author did succeed in doing so, thereby obtaining the structure for \(G\) explicitly.

MSC:

20C15 Ordinary representations and characters
20D15 Finite nilpotent groups, \(p\)-groups
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