In mathematics, in the field of group theory, an abnormal subgroup is a subgroup H of a group G such that for every x ∈ G, x lies in the subgroup generated by H and H x, where Hx 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.
Famous quotes containing the word abnormal:
“We urgently need a debate about the best ways of supporting families in modern America, without blinders that prevent us from seeing the full extent of dependence and interdependence in American life. As long as we pretend that only poor or abnormal families need outside assistance, we will shortchange poor families, overcompensate rich ones, and fail to come up with effective policies for helping families in the middle.”
—Stephanie Coontz (20th century)