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  "body": "\n## Definition\n\nLet $\\Gamma$ be a finite simple [[graph|00C0BR7D7H]] with vertex set $V$ and edge set $E$.\nThe *right-angled Artin group* $A(\\Gamma)$ is the group with [[presentation|0JGEQXQ4AC]]\n$$A(\\Gamma) = \\langle V \\mid uv = vu \\text{ for each edge } \\set{u, v} \\in E \\rangle.$$\nA group is a right-angled Artin group if it is isomorphic to $A(\\Gamma)$ for some finite simple graph $\\Gamma$.\n\n## Remarks\n\nExample: if $\\Gamma$ has $n$ vertices and no edges, $A(\\Gamma)$ is the [[free group|0KTW26C80J]] $F_n$; if $\\Gamma$ is complete, $A(\\Gamma)$ is the [[free abelian group|0WJX8MGG37]] $\\Z^n$.\nExample: if $\\Gamma$ is the path $a - b - c$, the relations say that $b$ commutes with $a$ and $c$, and $a, c$ generate a free group, so $A(\\Gamma) \\cong F_2 \\times \\Z$.\n\nNon-example: the braid group $B_3 = \\langle a, b \\mid aba = bab \\rangle$ is an Artin group but not a right-angled one.\nIts abelianisation is $\\Z$, since the relation forces $a = b$ there; the only right-angled Artin group with abelianisation $\\Z$ is $\\Z$ itself, and $B_3$ is not abelian.\n\nThe abelianisation of $A(\\Gamma)$ is $\\Z^{V}$.\nTwo right-angled Artin groups $A(\\Gamma)$ and $A(\\Gamma')$ are isomorphic if and only if the graphs $\\Gamma$ and $\\Gamma'$ are isomorphic.\nThey are also called graph groups or partially commutative groups.\n",
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