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