pronormal subgroup

A subgroup $H$ of a group $G$ is called a pronormal subgroupif for all $x\\in G$ the subgroups $H$ and $xHx^{-1}$ are conjugate in ${\\left\\langle H,xHx^{-1}\\right\\rangle}$ .

Some facts about pronormal subgroups:

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Normal subgroups are pronormal.
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Maximal subgroups are pronormal.
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Abnormal subgroups are pronormal.
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Sylow subgroups of finite groups are pronormal.
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The normalizer of a pronormal subgroup is abnormal.
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A pronormal subgroup is normal if and only if it is subnormal.

单词	PronormalSubgroup
释义	pronormal subgroup A subgroup $H$ of a group $G$ is called a pronormal subgroupif for all $x\\in G$ the subgroups $H$ and $xHx^{-1}$ are conjugate in ${\\left\\langle H,xHx^{-1}\\right\\rangle}$ . Some facts about pronormal subgroups: • Normal subgroups are pronormal. • Maximal subgroups are pronormal. • Abnormal subgroups are pronormal. • Sylow subgroups of finite groups are pronormal. • The normalizer of a pronormal subgroup is abnormal. • A pronormal subgroup is normal if and only if it is subnormal.
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