Normal series

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In mathematics, a normal series (also normal tower. subnormal series, subinvariant series, or just series) of a group G is a sequence of subgroups, each a normal subgroup of the next one. In a standard notation

There is no requirement made that A_i be a normal subgroup of G, only a normal subgroup of A_i+1. The quotient groups A_i+1/A_i are called the factor groups of the series.

If in addition each A_i is normal in G, then the series is called an invariant series (or a normal series, when this term is not used for the weaker sense).

A series with the additional property that A_i ≠ A_i+1 for all i is called a normal series without repetition; equivalently, each A_i is a proper, normal subgroup of A_i+1. The length of a normal series is the number of strict inclusions A_i < A_i+1. Equivalently, the length is the number of nontrivial factor groups. If the series has no repetition the length is just n.

Clearly, every (nontrivial) group has a normal series of length 1, namely . For simple groups this is the longest series possible.

A refinement of a normal series is another normal series containing each of the terms of the original series. Two normal series are said to be equivalent or isomorphic if there is a bijection between the sets of their factor groups such that the corresponding factor groups are isomorphic.

A composition series of a group is a normal series for which each of the A_i is a maximal normal subgroup of A_i+1. Equivalently, a composition series is a normal series for which each of the factor groups are simple.

A solvable group, or soluble group, is one with a normal series whose factor groups are all abelian.