An automorphism $\phi : X \rightleftarrows X$ of some [[algebra]]ic structure $X$ is an [[isomorphism]] from $X$ to itself. Note that the set of automorphisms $\mathrm{Aut}(X)$ forms a [[group]] under composition (called the **automorphism group** or **symmetric group** of $X$): - $f, g \in \mathrm{Aut}(X) \implies f \circ g \in \mathrm{Aut}(X)$ - $\mathrm{id}_{X} \in \mathrm{Aut}(X)$ - $f \in \mathrm{Aut}(X) \implies f^{-1} \in \mathrm{Aut}(X)$ In particular, for a finite $X$ of $N$ elements, the symmetry group is exactly the set of permutations of those elements. # sources [[1991PierceBasicCategoryTheory]] [Symmetric Group | Brilliant Math & Science Wiki](https://brilliant.org/wiki/symmetry-group/) [Symmetric group - Wikipedia](https://en.wikipedia.org/wiki/Symmetric_group)