If G is a

group, a series of

subgroups
G=G_{1}>G_{2}>...>G_{n}>G_{n+1}={1}

(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, G_{i} is normal in G.

The series is said to be __subnormal__ if for each i=1,...,n, G_{i+1} is normal in G_{i}. 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.