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Finite groups with few self-normalizing subgroups. (English) Zbl 1268.20022
The purpose of this paper is to describe the structure of finite groups which contain just one conjugate class of self-normalizing groups. If \(G\) is a group with such property, then there is a normal subgroup \(N\leq Z_\infty(G)\) of \(G\) such that \(G/N=UW\), where \(U\) is an elementary Abelian normal Sylow \(p\)-group of \(G/N\), \(W\) is a nilpotent Hall \(p'\)-subgroup of \(G/N\) and \(W\) acts irreducibly on \(U\).
It is clear that \(G\) is soluble and \(Z_\infty(G)H\) is a Carter subgroup of \(G\).
MSC:
20D25 Special subgroups (Frattini, Fitting, etc.)
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20E34 General structure theorems for groups
20E45 Conjugacy classes for groups
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