# Composition Series in a Group

Definition: Let $G$ be a group. A Composition Series for $G$ is a (finite) chain of successive subgroups of $G$, denoted by $\{ e \} = G_0 \leq G_1 \leq ... \leq G_n = G$ with the following properties:1) $G_i$ is a normal subgroup of $G_{i+1}$ for all $0 \leq i \leq n-1$.2) $G_{i+1}/G_i$ is a simple group for all $0 \leq i \leq n-1$.The Length of the composition series is the number $n$, and the Composition Factors of the composition series are the quotient groups $G_{i+1}/G_i$. |

*Recall that a group $G$ is said to be a simple group if it has no nontrivial proper normal subgroups. Thus condition (2) requires each of the composition factors $G_{i+1}/G_i$ to have no nontrivial proper normal subgroups.*

For example, consider $G = \mathbb{Z}_{12}$, the cyclic group of order $12$, and write $\mathbb{Z}_{12} = \langle a \rangle$ so that $a^{12} = 1$. Recall that every subgroup of a cyclic group is cyclic. Thus, all of the subgroups of $G = \mathbb{Z}_{12}$ are:

(1)Then $\mathbb{Z}_1, \mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_4$ and $\mathbb{Z}_6$ are subgroups of $\mathbb{Z}_{12}$. Since every cyclic group is abelian, we have these each of these groups are normal subgroups. Consider the following chain of subgroups:

(2)The three series above are all composition series for $G = \mathbb{Z}_{12}$. Indeed, each subgroup is normal in the next since all of the groups are abelian. Moreover, the composition factors in all three composition series are isomorphic to either $\mathbb{Z}_2$ or $\mathbb{Z}_3$ - both of which are simple groups as they have no nontrivial proper subgroups - and hence, no nontrivial proper normal subgroups.