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  text: 'Let $G$ be a [-3gd], [3t9 acting] on the set $X$.\nThen the [group_orbit orbits] of $X$ under $G$ form a [set_partition partition] of $X$.\n\n# Proof\n\nWe need to show that every element of $X$ is in an orbit, and that if $x \\in X$ lies in two orbits then they are the same orbit.\n\nCertainly $x \\in X$ lies in an orbit: it lies in the orbit $\\mathrm{Orb}_G(x)$, since $e(x) = x$ where $e$ is the identity of $G$.\n(This follows by the definition of an action.)\n\nSuppose $x$ lies in both $\\mathrm{Orb}_G(a)$ and $\\mathrm{Orb}_G(b)$, where $a, b \\in X$.\nThen $g(a) = h(b) = x$ for some $g, h \\in G$.\nThis tells us that $h^{-1}g(a) = b$, so in fact $\\mathrm{Orb}_G(a) = \\mathrm{Orb}_G(b)$; it is an exercise to prove this formally.\n\n%%hidden(Show solution):\nIndeed, if $r \\in \\mathrm{Orb}_G(b)$, then $r = k(b)$, say, some $k \\in G$.\nThen $r = k(h^{-1}g(a)) = kh^{-1}g(a)$, so $r \\in \\mathrm{Orb}_G(a)$.\n\nConversely, if $r \\in \\mathrm{Orb}_G(a)$, then $r = m(b)$, say, some $m \\in G$.\nThen $r = m(g^{-1}h(b)) = m g^{-1} h (b)$, so $r \\in \\mathrm{Orb}_G(b)$.\n%%\n\n',
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