Abstract
The following theorems are proved about the Frattini-free componentG Φ of a soluble stable ℜ-group: a) If it has a normal subgroupN with nilpotent quotientG Φ/N, then there is a nilpotent subgroupH ofG Φ withG Φ=NH. b) It has Carter subgroups; if the group is small, they are all conjugate. c) Nilpotency modulo a suitable Frattini-subgroup (to be defined) implies nilpotency. The last result makes use of a new structure theorem for the centre of the derivative of the Frattini-free component of a centreless soluble ℜ-group.
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Wagner, F.O. Nilpotent complements and Carter subgroups in stable ℜ-groups. Arch Math Logic 33, 23–34 (1994). https://doi.org/10.1007/BF01275468
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DOI: https://doi.org/10.1007/BF01275468