Abnormal subgroup

In mathematics, specifically group theory, an abnormal subgroup is a subgroupH of a groupG such that for all x in G, x lies in the subgroup generated by H and H, where H denotes the conjugate subgroupxHx.

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 , and every weakly abnormal subgroup is a self-normalizing subgroup.
• Every abnormal subgroup is a pronormal subgroup, and hence a , a paranormal subgroup, and a polynormal subgroup.