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 {\displaystyle H} is k {\displaystyle k} -subnormal in G {\displaystyle G} if there are subgroups

H = H 0 , H 1 , H 2 , … , H k = G {\displaystyle H=H_{0},H_{1},H_{2},\ldots ,H_{k}=G}

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

A subnormal subgroup is a subgroup that is k {\displaystyle k} -subnormal for some positive integer k {\displaystyle 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.