Subnormal subgroup

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.

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

:H=H_0,H_1,H_2,\ldots, H_k=G

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 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.

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