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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] with identity element $e$, and let $a \\in G$.\nFor an integer $n$, define $a^n$ as follows: $a^0 = e$, $a^{n+1} = a^n a$ for $n \\geq 0$, and $a^{-n} = (a^{-1})^n$ for $n \u003e 0$.\nThen $a^{m+n} = a^m a^n$ and $(a^m)^n = a^{mn}$ for all integers $m$ and $n$.\n\n## Remarks\n\nThe law $(ab)^n = a^n b^n$ is true for all $n$ when $ab = ba$, but it can fail when $a$ and $b$ do not commute.\nIn additive notation the laws are $(m + n)a = ma + na$ and $n(ma) = (nm)a$.\nThe first law shows that $n \\mapsto a^n$ is a group homomorphism from $(\\mathbb{Z}, +)$ to $G$.\n\n## Explanations\n\n### L1 human\n\nThe powers of an element of a group follow the rules that you know for the powers of a nonzero number.\nA negative power means a power of the inverse, so $a^{-2} = a^{-1} a^{-1}$.\nFor example, $a^3 a^{-1} = a a a a^{-1} = a^2$.\n",
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    "name": "Laws of exponents in a group",
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  "msc": [
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  "name": "Laws of exponents in a group",
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