Definition:Automorphism Group/Group

Definition
Let $\struct {S, *}$ be an algebraic structure.

Let $\mathbb S$ be the set of automorphisms of $S$.

Then the algebraic structure $\struct {\mathbb S, \circ}$, where $\circ$ denotes composition of mappings, is called the automorphism group of $S$.

The structure $\struct {S, *}$ is usually a group. However, this is not necessary for this definition to be valid.

The automorphism group of $S$ is denoted on as $\Aut S$.

Also known as
The automorphism group is also known as the group of automorphisms.

The automorphism group of $S$ can be found denoted in a number of ways, for example:
 * $\map {\mathscr A} S$
 * $\map A S$

Also see

 * Automorphism Group is Subgroup of Symmetric Group, where it is also demonstrated that $\Aut S$ is actually a group.