---
tag: 0RFP7F3SEW
kind: definition
name: Composition factors
msc: ["20E15", 20D30]
defines: 0PHTC93M58
uses: [0NB8VPB93V]
sources:
  - url: https://en.wikipedia.org/wiki/Composition_series
    locator: For groups
---

## Definition

Let $G$ be a [[group|G5S1M9]] with a [[composition series|0XV2A49X6Q]] $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$.
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|0E088ZP533]] 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.