Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / B / B2 / a

A composition series is a finite chain $1=G_0⊲G_1⊲\cdots ⊲G_r=G$ in which each inclusion is proper, each subgroup is normal in the next, and every quotient $G_i/G_{i-1}$ is a nontrivial simple group. These quotients are the composition factors.

For a finite group, use induction on its order. The trivial group has the empty series. Otherwise choose a maximal proper normal subgroup $N$; such a subgroup exists because there are finitely many subgroups and $1$ is proper. The correspondence theorem says normal subgroups of $G/N$ correspond to normal subgroups of $G$ containing $N$, so maximality makes $G/N$ simple. Inductively construct a composition series of the smaller group $N$ and append $G$. This proves existence.

A nonabelian example with abelian factors isBoth factors are simple because their orders are prime.