Composition factors

Definition

Let $G$ be a group with a composition series $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$. The composition factors of $G$ are the simple groups $G_i / G_{i-1}$ for $i = 1, \dots, n$, counted with multiplicity. The number $n$ is the composition length of $G$.

Remarks

The Jordan–Hölder theorem shows that the composition factors do not depend on the series, up to order and isomorphism. The composition factors of $\mathbb{Z}/12\mathbb{Z}$ are $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$. The composition factors of $S_4$ are $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$. Groups that are not isomorphic can have the same composition factors, for example $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

Explanations

L1 human

The composition factors are the simple pieces of a group. They are like the prime factors of an integer, counted with multiplicity. But unlike for integers, the pieces do not determine the group.