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

$H=H_0,H_1,H_2,\dots ,H_k=G$

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.

See also

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