Abnormal subgroup explained

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^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

• Fattahi . Abiabdollah . Groups with only normal and abnormal subgroups . Journal of Algebra . 28 . 1 . 15–19 . Elsevier . January 1974 . 10.1016/0021-8693(74)90019-2. free .
• Zhang . Q. H. . Finite groups with only seminormal and abnormal subgroups . J. Math. Study . 29 . 4 . 10–15 . 1996.
• Zhang . Q. H. . Finite groups with only ss-quasinormal and abnormal subgroups . Northeast. Math. J. . 14 . 1 . 41–46 . 1998 .
• Zhang . Q. H. . s-Semipermutability and abnormality in finite groups . Comm. Algebra . 27 . 9 . 4515–4524 . 1999 . 10.1080/00927879908826711.