Definition:Group of Automorphisms

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Definition

Let $\left({S, *}\right)$ be an algebraic structure.

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

Then the algebraic structure $\left({\mathbb S, \circ}\right)$, where $\circ$ denotes composition of mappings, is called the group of automorphisms of $S$.

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

The group of automorphisms of $S$ is often denoted $\operatorname{Aut} \left({S}\right)$ or $\mathscr A \left({S}\right)$.

Group of Automorphisms is Subgroup of Group of Permutations, where it is also demonstrated that $\operatorname{Aut} \left({S}\right)$ is actually a group.