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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] with identity element $e$, and let $a, b \\in G$.\nIf $ab = e$, then $b = a^{-1}$.\n\n## Remarks\n\nIn particular, each element of a group has exactly one [[inverse|0ZNZH0EXFX]].\nThe same argument works in a [[monoid|0N6R902FG1]]: if $ba = e$ and $ac = e$, then $b = b(ac) = (ba)c = c$.\nTwo consequences are $(a^{-1})^{-1} = a$ and $(ab)^{-1} = b^{-1} a^{-1}$.\n\n## Explanations\n\n### L1 human\n\nIn a group, you only need to check one equation to know that $b$ is the inverse of $a$.\nMultiply $ab = e$ on the left by $a^{-1}$, and you get $b = a^{-1}$.\nThus the notation $a^{-1}$ makes sense: there is only one inverse.\n",
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