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| abnormal subgroup ： ウィキペディア英語版 |
abnormal subgroup
In mathematics, in the field of group theory, an abnormal subgroup is a subgroup ''H'' of a group ''G'' such that for every ''x'' ∈ ''G'', ''x'' lies in the subgroup generated by ''H'' and ''H'' ''x'', where ''H''''x'' denotes the conjugate subgroup ''xHx''''-1''.
Here are some facts relating abnormality to other subgroup properties:
* Every abnormal subgroup is a self-normalizing subgroup, as well as a contranormal subgroup.
* The only normal subgroup that is also abnormal is the whole group.
* Every abnormal subgroup is a weakly abnormal subgroup, and every weakly abnormal subgroup is a self-normalizing subgroup.
* Every abnormal subgroup is a pronormal subgroup, and hence a weakly pronormal subgroup, a paranormal subgroup, and a polynormal subgroup.
==References==
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抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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