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  text: 'The dihedral group $D_{2n}$ is the group of symmetries of the $n$-vertex [-regular_polygon].\n\n# Presentation\nThe dihedral groups have very simple [group_presentation presentations]: $$D_{2n} \\cong \\langle a, b \\mid a^n, b^2, b a b^{-1} = a^{-1} \\rangle$$\nThe element $a$ represents a rotation, and the element $b$ represents a reflection in any fixed axis.\n[todo: picture]\n\n# Properties\n\n- The dihedral groups $D_{2n}$ are all non-abelian for $n > 2$. ([4d0 Proof.])\n- The dihedral group $D_{2n}$ is a [-subgroup] of the [-497] $S_n$, generated by the elements $a = (123 \\dots n)$ and $b = (2, n)(3, n-1) \\dots (\\frac{n}{2}+1, \\frac{n}{2}+3)$ if $n$ is even, $b = (2, n)(3, n-1)\\dots(\\frac{n-1}{2}, \\frac{n+1}{2})$ if $n$ is odd.\n\n# Examples\n\n## $D_6$, the group of symmetries of the triangle\n\n[todo: diagram]\n[todo: list the elements and Cayley table]\n\n# Infinite dihedral group\n\nThe infinite dihedral group has presentation $\\langle a, b \\mid b^2, b a b^{-1} = a^{-1} \\rangle$.\nIt is the "infinite-sided" version of the finite $D_{2n}$.\n\nWe may view the infinite dihedral group as being the subgroup of the group of [homeomorphism homeomorphisms] of $\\mathbb{R}^2$ generated by a reflection in the line $x=0$ and a translation to the right by one unit.\nThe translation is playing the role of a rotation in the finite $D_{2n}$.\n\n[todo: this section]',
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