abnormal subgroup

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^{{{space|hair}}x}, where H^{{{space|hair}}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

{{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