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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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