Contranormal subgroup

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In mathematics, in the field of group theory, a contranormal subgroup is a subgroup whose normal closure in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is the whole group.

Some facts:

• Every subgroup of a finite group is a contranormal subgroup of a subnormal subgroup. In general, every subgroup of a group is a contranormal subgroup of a descendant subgroup.
• Every contranormal subgroup which is also a normal subgroup, must be the whole group.
• Every abnormal subgroup is contranormal.