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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\n### L2 agent\n\nThe definition of a group asks for an inverse $b$ with $ab = e$ and $ba = e$. This result shows that one of the two equations is enough, and that the inverse is unique.\n\nThe proof uses only associativity and the identity:\n\n1. Suppose that $ab = e$.\n2. Multiply on the left by $a^{-1}$: $a^{-1}(ab) = a^{-1}e$.\n3. By associativity, the left side is $(a^{-1}a)b = eb = b$.\n4. The right side is $a^{-1}$. Thus $b = a^{-1}$.\n\nA worked example in $\\mathbb{Z}/7\\mathbb{Z}$ under multiplication of the units:\n\n1. We look for the inverse of $3$.\n2. We have $3 \\cdot 5 = 15 = 2 \\cdot 7 + 1$, so $3 \\cdot 5 = 1$.\n3. By the result, $5$ is the only inverse of $3$. There is no need to check $5 \\cdot 3$.\n\nTwo standard consequences follow at once. First, $(a^{-1})^{-1} = a$, because $a^{-1}a = e$. Second, $(ab)^{-1} = b^{-1}a^{-1}$, because $(ab)(b^{-1}a^{-1}) = e$.\n\nA common error is to write $(ab)^{-1} = a^{-1}b^{-1}$. This is false when $a$ and $b$ do not commute. In $S_3$, take $a = (1\\,2)$ and $b = (1\\,3)$: then $a^{-1}b^{-1} = (1\\,2)(1\\,3)$, but $(ab)^{-1} = (1\\,3)(1\\,2)$, and these differ.\n\nThe step that needs care is step 2. It uses the inverse $a^{-1}$, so the argument is for groups. In a monoid an element can have a right inverse and no left inverse.\n",
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        "adapter": "Math2.Match.inv_unique",
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    "name": "Uniqueness of inverses",
    "role": "proposition",
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        "locator": "Definitions and basic properties",
        "url": "https://en.wikipedia.org/wiki/Inverse_element"
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  "markdown": "https://tamnd.github.io/math2/node/0D9NDGE9KP.md",
  "msc": [
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  "name": "Uniqueness of inverses",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/0D9/0D9NDGE9KP.md",
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  "url": "https://tamnd.github.io/math2/wiki/uniqueness-of-inverses/",
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