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  "body": "\n## Example\n\nThe [[dihedral group|0JZ27DMJ5D]] $D_4$ is the group of the symmetries of a square, and it has order $8$.\nLet $r$ be the rotation by $90$ degrees and let $s$ be a reflection in a line of symmetry.\nThen $D_4 = \\{e, r, r^2, r^3, s, rs, r^2s, r^3s\\}$, with $r^4 = e$, $s^2 = e$ and $srs^{-1} = r^{-1}$.\nThe elements $r$ and $r^3$ have order $4$, and the other five elements different from $e$ have order $2$.\nThe [[centre|0DZ93493VN]] of $D_4$ is $\\{e, r^2\\}$, because the rotation by $180$ degrees commutes with each symmetry.\nThe rotation subgroup $\\langle r \\rangle$ is [[normal|0CB0DX5BVV]] of index $2$.\nThe subgroup $\\{e, s\\}$ is not normal, because $r s r^{-1} = r^2 s$.\nNumbering the vertices $1, 2, 3, 4$ gives an injective homomorphism from $D_4$ into $S_4$, and its image is a Sylow $2$-subgroup of $S_4$.\n",
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