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text: 'Cayley's Theorem states that every group $G$ appears as a certain subgroup of the [-497] $\\mathrm{Sym}(G)$ on the [-3gz] of $G$.\n\n# Formal statement\n\nLet $G$ be a group.\nThen $G$ is [49x isomorphic] to a subgroup of $\\mathrm{Sym}(G)$.\n\n# Proof\n\nConsider the left regular [3t9 action] of $G$ on $G$: that is, the function $G \\times G \\to G$ given by $(g, h) \\mapsto gh$.\nThis [49c induces a homomorphism] $\\Phi: G \\to \\mathrm{Sym}(G)$ given by [-currying]: $g \\mapsto (h \\mapsto gh)$.\n\nNow the following are equivalent:\n\n- $g \\in \\mathrm{ker}(\\Phi)$ the [49y kernel] of $\\Phi$\n- $(h \\mapsto gh)$ is the identity map\n- $gh = h$ for all $h$\n- $g$ is the identity of $G$\n\nTherefore the kernel of the homomorphism is trivial, so it is injective.\nIt is therefore bijective onto its [3lh image], and hence an isomorphism onto its image.\n\nSince [4b4 the image of a group under a homomorphism is a subgroup of the codomain of the homomorphism], we have shown that $G$ is isomorphic to a subgroup of $\\mathrm{Sym}(G)$.',
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