If G is a group, a series of subgroups


(where G>H means H is a subgroup of G and {1} denotes the trivial group) is said to be normal if for each i=1,...,n,n+1, Gi is normal in G.

The series is said to be subnormal if for each i=1,...,n, Gi+1 is normal in Gi. So a normal series is a subnormal series, but the converse is not true in general.

See also composition series, Jordan-Holder theorem, soluble group.