In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G.

if there are subgroups

.

. Some facts about subnormal subgroups:

• A 1-subnormal subgroup is a proper normal subgroup (and vice versa).

• A finitely generated group is nilpotent if and only if each of its subgroups is subnormal.

• Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal.

• Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal.

• Every 2-subnormal subgroup is a conjugate-permutable subgroup.

The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality.

If every subnormal subgroup of G is normal in G, then G is called a T-group.

See also

• Characteristic subgroup

• Normal core

• Normal closure

• Ascendant subgroup

• Descendant subgroup

• Serial subgroup

References