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.

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• 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.

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