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text: 'Two elements $x, y$ of a [-3gd] $G$ are *conjugate* if there is some $h \\in G$ such that $hxh^{-1} = y$.\n\n# Conjugacy as "changing the worldview"\n\nConjugating by $h$ is equivalent to "viewing the world through $h$'s eyes".\nThis is most easily demonstrated in the [-497], where it [4bh is a fact] that if $$\\sigma = (a_{11} a_{12} \\dots a_{1 n_1})(a_{21} \\dots a_{2 n_2}) \\dots (a_{k 1} a_{k 2} \\dots a_{k n_k})$$\nand $\\tau \\in S_n$, then $$\\tau \\sigma \\tau^{-1} = (\\tau(a_{11}) \\tau(a_{12}) \\dots \\tau(a_{1 n_1}))(\\tau(a_{21}) \\dots \\tau(a_{2 n_2})) \\dots (\\tau(a_{k 1}) \\tau(a_{k 2}) \\dots \\tau(a_{k n_k}))$$\n\nThat is, conjugating by $\\tau$ has "caused us to view $\\sigma$ from the point of view of $\\tau$".\n\nSimilarly, in the [-4cy] $D_{2n}$ on $n$ vertices, conjugation of the rotation by a reflection yields the inverse of the rotation: it is "the rotation, but viewed as acting on the reflected polygon".\nEquivalently, if the polygon is sitting on a glass table, conjugating the rotation by a reflection makes the rotation act "as if we had moved our head under the table to look upwards first".\n\nIn general, if $G$ is a group which [3t9 acts] as (some of) the symmetries of a certain object $X$ %%note:Which [49b we can always view as being the case].%% then conjugation of $g \\in G$ by $h \\in G$ produces a symmetry $hgh^{-1}$ which acts in the same way as $g$ does, but on a copy of $X$ which has already been permuted by $h$.\n\n# Closure under conjugation\n\nIf a subgroup $H$ of $G$ is closed under conjugation by elements of $G$, then $H$ is a [-4h6].\nThe concept of a normal subgroup is extremely important in group theory.\n\n%%%knows-requisite([3t9]):\n# Conjugation action\n\nConjugation forms a [3t9 action].\nFormally, let $G$ act on itself: $\\rho: G \\times G \\to G$, with $\\rho(g, k) = g k g^{-1}$.\nIt is an exercise to show that this is indeed an action.\n%%hidden(Show solution):\nWe need to show that the identity acts trivially, and that products may be broken up to act individually.\n\n- $\\rho(gh, k) = (gh)k(gh)^{-1} = ghkh^{-1}g^{-1} = g \\rho(h, k) g^{-1} = \\rho(g, \\rho(h, k))$;\n- $\\rho(e, k) = eke^{-1} = k$.\n%%\n\nThe [group_stabiliser stabiliser] of this action, $\\mathrm{Stab}_G(g)$ for some fixed $g \\in G$, is the set of all elements such that $kgk^{-1} = g$: that is, such that $kg = gk$.\nEquivalently, it is the [group_centraliser centraliser] of $g$ in $G$: it is the subgroup of all elements which commute with $G$.\n\nThe [group_orbit orbit] of the action, $\\mathrm{Orb}_G(g)$ for some fixed $g \\in G$, is the [-4bj] of $g$ in $G$.\nBy the [-4l8], this immediately gives that the size of a conjugacy class divides the [3gg order] of the parent group.\n%%%',
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