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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 chain of subgroups of the group, each one normal in the next, beginning at H and ending at G.

In notation, H is k-subnormal in G if there are subgroups

of G such that H_i is normal in H_{i + 1} for each i.

A subnormal subgroup is a subgroup that is k-subnormal for some positive integer k Some facts about subnormal subgroups:

• A 1-subnormal subgroup is a normal subgroup (and vice versa).
• A finite group is a nilpotent group if and only if every subgroup of it 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, in fact, normal. In particular, every 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. In fact, the relation of subnormality can be defined as the transitive closure of the relation of normality.

[edit] See also

• Normal subgroup
• Characteristic subgroup
• Normal core
• Normal closure
• Ascendant subgroup
• Descendant subgroup
• Serial subgroup