View source for Pronormal subgroup

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A subgroup $H$ of a [[Group|group]] $G$ satisfying the following condition: If $K$ is a subgroup in $G$ conjugate with $H$, then $K$ is conjugate with $H$ in the subgroup generated by $H$ and $K$ (cf. [[Conjugate elements|Conjugate elements]]). Sylow subgroups in finite groups, as well as Hall and Carter subgroups in finite solvable groups, are pronormal (cf. [[Sylow subgroup|Sylow subgroup]]; [[Hall subgroup|Hall subgroup]]; [[Carter subgroup|Carter subgroup]]). The concept of a pronormal subgroup is closely connected with that of an [[Abnormal subgroup|abnormal subgroup]]. Every abnormal subgroup is pronormal, and the normalizer of a pronormal subgroup (cf. [[Normalizer of a subset|Normalizer of a subset]]) is abnormal.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Shemetkov,   "Formations of finite groups" , Moscow  (1978)  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,   "A course in the theory of groups" , Springer  (1982)</TD></TR></table>