---
tag: WYGRXCG2RN
kind: concept
name: Group representation
msc: [20C15, 20C05]
sources:
  - url: https://en.wikipedia.org/wiki/Group_representation
same_as:
  wikidata: Q1055807
  wikipedia: Group representation
---

## Summary

A representation of a [[group|G5S1M9]] $G$ on a [[vector space|000AYPA9AJ]] $V$ is a [[group homomorphism|0JYTG5K9Y4]] $G \to GL(V)$.
Here $GL(V)$ is the group of invertible [[linear maps|0NRYHJPQG2]] $V \to V$.
It lets each element of $G$ act on $V$ as a linear transformation, compatibly with the group operation.
Representations turn questions about groups into questions of linear algebra, where matrices, traces and eigenvalues are available.
They are central in group theory, number theory, physics and chemistry.

## Explanations

### L1 human

A representation lets a group act by matrices.
Each group element becomes an invertible matrix, and multiplying group elements corresponds to multiplying the matrices.
For example, the cyclic group of order $4$ can be represented on the plane $\R^2$ by sending a generator to the rotation by $90$ degrees.
Then the four group elements become the four rotations by $0$, $90$, $180$ and $270$ degrees.