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Finite groups in which all non-normal subgroups have the same order. (Italian) Zbl 1097.20020

R. Brandl [Commun. Algebra 23, No. 6, 2091-2098 (1995; Zbl 0830.20042)] has asked: given a finite group determine its non-normal subgroups.
Let \(G\) be a non-Abelian and non-Hamiltonian finite group and \(n\) an integer \(\geq 2\). A group \(G\) is in the class \(S(n)\) if all its non-normal subgroups have order \(n\). Given a prime number \(p\), the author determines all \(p\)-groups in: (1) \(S(p)\); (2) \(S(p^i)\) with \(i>1\) and \(p\geq 3\). Furthermore, all groups of exponent \(4\) in the class \(S(4)\) are presented.

MSC:

20D30 Series and lattices of subgroups
20D25 Special subgroups (Frattini, Fitting, etc.)
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
20E34 General structure theorems for groups

Citations:

Zbl 0830.20042
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