---
tag: 0ZG3Z5JZ85
kind: concept
name: Symmetry group
msc: [20B25]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetry_group
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Symmetry_group
    locator: Symmetry groups in general
same_as:
  wikidata: Q902019
  wikipedia: Symmetry group
---

## Summary

The symmetry group of an object is the group of all transformations of the ambient space that map the object onto itself. These transformations must keep the relevant structure of the object.
The operation is composition of transformations.
It is often written $\operatorname{Sym}(X)$ for an object $X$.

## Remarks

The symmetry group of a regular $n$-gon in the plane is the [[dihedral group|0JZ27DMJ5D]] of order $2n$.
The rotations of a regular tetrahedron form a group [[isomorphic|08MG7H8JW5]] to the [[alternating group|0Z5905Q73D]] $A_4$.
The symmetry group of an object is a [[subgroup|0X1JXT47GA]] of the group of all transformations of the ambient space.

## Explanations

### L1 human

A symmetry of an object is a motion that leaves the object looking the same.
Two symmetries done one after the other give another symmetry, and each symmetry can be undone.
So the symmetries of an object form a group, and this group measures how symmetric the object is.