In group theory, a composition series is a special kind of subnormal series which gives a decomposition of a group into simple groups.

Specifically, if G is a group, then a composition series for G is a sequence


(where {1} denotes the trivial group and G>H means "H is a subgroup of G") such that for i=1,...,n,

  1. Gi+1 is a normal subgroup of Gi and
  2. the quotient group Gi/Gi+1 is a simple group.

These quotient groups are called composition factors for G.

It can be shown, using induction on the size of the group and the isomorphism theorems, that every finite group has a composition series. In general, the composition series will not be unique, but the Jordan-Holder theorem states that the composition factors will be unique for each group (counting mulitplicity).

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