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 of order $2n$. The rotations of a regular tetrahedron form a group isomorphic to the alternating group $A_4$. The symmetry group of an object is a subgroup 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.