Tag Info

Use with the (group-theory) tag. Groups describe the symmetries of an object through their actions on the object. For example, dihedral groups of order $2n$ act on regular $n$-gons, $S_n$ acts on the numbers $\{1, 2, \ldots, n\}$ and the Rubik's cube group acts on Rubik's cube.

Groups describe the symmetries of an object through their actions on the object. For example, dihedral groups of order $2n$ act on regular $n$-gons, $S_n$ acts on the numbers $\{1, 2, \ldots, n\}$ and the Rubik's cube group acts on Rubik's cube.

When describing a group action, it should be clear whether the action is a left action or a right action. A right action is a function,

$\cdot : X\times G\rightarrow X$

such that $x\cdot 1=x$ for all $x\in X$ and such that $(x\cdot g)\cdot h=x\cdot (gh)$. These conditions mean the action of the group makes sense; that the action is compatible with the group.

A left action is defined analogously.

If $G$ acts on $X$ then there exists a homomorphism of groups $G\rightarrow \operatorname{Aut}(X)$. This is of interest when $X$ is a group too, and allows us to construct semidirect products of groups.

For more details, see Wikipedia.