= Symmetry group
{title2=$\operatorname{Aut}(X)$}
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A symmetry group collects <bijections> of an object to itself that preserve a specified structure in both directions, with <function composition> as the operation. For a geometric object one may require <Euclidean distances> to be preserved, giving an <isometry group>; for a <graph>, one preserves adjacency, giving a <graph automorphism> <group>. For an unstructured <finite set>, every <permutation> is allowed. \b[The structure to be preserved is part of the definition.]
For a <square> with its usual geometric structure, the <symmetry group> consists of four <rotations> and four <reflections>. These form the <dihedral group> of <group order> eight. The existing <group action> example shows how its eight elements move the <vertices of a polygon>; changing the object to its edges or a marked configuration changes the <group action> being studied.<Dihedral Groups by Keith Conrad>[§§1–2, Theorem 2.5, printed pp. 1–4: rigid motions, rotations, reflections and dihedral order conventions]{ref}
Specifying the preserved data makes it possible to test a proposed symmetry exactly. After the preserved data are declared, <function composition>, the <identity map>, and <inverse maps> automatically preserve them too. <Symmetry transformations> therefore form a <group>, and questions about equivalent configurations become questions about <group orbits>, <stabilizer subgroups>, and <group homomorphisms>. This supplies reusable mathematics for comparing many different kinds of objects.
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