In mathematics, specifically group theory, an '''abnormal subgroup''' is a subgroup ''H'' of a group ''G'' such that for all ''x'' in ''G'', ''x'' lies in the subgroup generated by ''H'' and ''H''<sup>{{space|hair}}''x''</sup>, where ''H''<sup>{{space|hair}}''x''</sup> denotes the conjugate subgroup ''xHx''<sup>−1</sup>.
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== * {{cite journal | last = Fattahi | first = Abiabdollah | title = Groups with only normal and abnormal subgroups | journal = Journal of Algebra | volume = 28 | issue = 1 | pages = 15–19 | publisher = Elsevier | date = January 1974 | doi = 10.1016/0021-8693(74)90019-2| doi-access = free }} * {{cite journal | last = Zhang | first = Q. H. | title = Finite groups with only seminormal and abnormal subgroups | journal = J. Math. Study | volume = 29 | issue = 4 | pages = 10–15 | year = 1996}} * {{cite journal | last = Zhang | first = Q. H. | title = Finite groups with only ss-quasinormal and abnormal subgroups | journal = Northeast. Math. J. | volume = 14 | issue = 1 | pages = 41–46 | year = 1998 }} * {{cite journal | last = Zhang | first = Q. H. | title = s-Semipermutability and abnormality in finite groups | journal = Comm. Algebra | volume = 27 | issue = 9 | pages = 4515–4524 | year = 1999 | doi=10.1080/00927879908826711}}
Category:Subgroup properties
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