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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $a, b \\in G$.\nThe element $b$ is conjugate to $a$ when there is $g \\in G$ with $b = gag^{-1}$.\nThe element $gag^{-1}$ is called the conjugate of $a$ by $g$.\n\n## Remarks\n\nConjugacy is an equivalence relation on $G$; its classes are the conjugacy classes.\nIn an [[abelian group|082GHNPJ53]], $gag^{-1} = a$, so each element is conjugate only to itself.\nConjugate elements have the same [[order|0WE2NHZJ7B]].\nIn the group of invertible matrices, conjugate matrices are similar matrices.\n\n## Explanations\n\n### L1 human\n\nConjugating $a$ by $g$ means: first undo $g$, then do $a$, then do $g$ again (reading the product from right to left).\nThe result does the same thing as $a$, but in a place that $g$ has moved.\nFor example, two rotations of a cube by the same angle about different axes are conjugate.\n",
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