= Conjugacy class
{title2=$\operatorname{Cl}_G(x)$}
{wiki}
For an element $x$ of a <group> $G$, its <conjugacy class> is its <orbit of a group action> under the <conjugation action>:
$$
\operatorname{Cl}_G(x)=\{gxg^{-1}:g\in G\}.
$$
Thus <conjugate group elements> describe the same operation after a change of labels. In a <symmetric group>, <conjugation> renames the entries of every <permutation cycle>:
$$
g(a_1\,a_2\,\ldots\,a_k)g^{-1}=(g(a_1)\,g(a_2)\,\ldots\,g(a_k)).
$$
For example, with $g=(123)$ and the <transposition> $t=(12)$, this gives $gtg^{-1}=(g(1)\,g(2))=(23)$. Products use <function composition> of <permutations>, with the rightmost factor applied first. The complete six-element example is <conjugacy classes of the symmetric group on three letters>.
The <conjugacy classes> form a <partition of a set> of $G$, since <conjugacy> is an <equivalence relation>. In a <symmetric group>, two <permutations> lie in the same <conjugacy class> precisely when they have the same <cycle type>: renaming preserves the lengths of <permutation cycles>, and matching equal-length <permutation cycles> constructs a renaming in the other direction. This compresses many labelled <permutations> into a few structural types. See https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/ae501fc20d40fdb32f9e1edfa64da8f3_MIT18_703S13_pra_l_6.pdf[McKernan, Conjugation in Sn, lemma 6.2 and proposition 6.6].
The <stabilizer subgroup> of $x$ under the <conjugation action> is its <centralizer> $C_G(x)$. Consequently the <orbit-stabilizer theorem> gives
$$
|\operatorname{Cl}_G(x)|=[G:C_G(x)],\qquad |\operatorname{Cl}_G(x)|=\frac{|G|}{|C_G(x)|}\quad\text{if }G\text{ is finite}.
$$
The counting mechanism is the <bijection> $gC_G(x)\mapsto gxg^{-1}$ from left <cosets> onto the <conjugacy class>: two choices of $g$ give the same result exactly when they differ on the right by an element of the <centralizer>. A large <centralizer> means many choices of labels leave the operation unchanged, hence a small <conjugacy class>. A <conjugacy class> has size one precisely when its element belongs to the <center of a group>. The <class equation> counts these <orbits of a group action> together.
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