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  "body": "\n## Definition\n\nLet $n \\geq 3$ be an integer.\nLet $P$ be a regular $n$-gon in the plane with centre $0$.\nThe dihedral group $D_n$ is the group of all isometries of the plane that map $P$ onto itself, with composition as the operation.\nLet $r$ be the rotation by the angle $2\\pi/n$ about $0$, and let $s$ be a reflection in a line of symmetry of the polygon.\n\n## Remarks\n\nThe elements of $D_n$ are the rotations $e, r, \\dots, r^{n-1}$ and the reflections $s, rs, \\dots, r^{n-1}s$.\nThe relations $r^n = e$, $s^2 = e$ and $srs^{-1} = r^{-1}$ hold, and these relations give a presentation of $D_n$.\nIn abstract algebra, many authors write $D_{2n}$ for this group, so the index is the order and not the number of sides.\nThe group $D_n$ is not abelian, and it is a [[semidirect product|0H3WZXCZNF]] of $\\langle r \\rangle$ and $\\langle s \\rangle$.\n\n## Explanations\n\n### L1 human\n\nTake a regular polygon and look at all the ways to pick it up and put it back in the same outline.\nYou can turn it, and you can flip it over.\nThese motions form the dihedral group, and turning after a flip is not the same as flipping after a turn.\n",
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