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For a finite Galois extension $L/K$ and a conjugacy class $C$ in its Galois group, the unramified primes with Frobenius class $C$ have density $|C|/|\mathrm{Gal}(L/K)|$.

A class function on a group is a function constant on every conjugacy class. Equivalently, $f(hg{h}^{-1})=f(g)$ for all group elements $g,h$.

The alternating conjugacy class splitting criterion gives seven conjugacy classes in $A_6$. Their cycle types and sizes are $1^6:1$, $2^2{1}^2:45$, $31^3:40$, $3^2:40$, $42:90$, and two $51$ classes of size 72. Representatives of the split classes are a 5-cycle and its square. Their sizes sum to 360. A normal subgroup must be a union of these classes containing the identity, so these explicit sizes can prove simplicity using Lagrange's theorem.

If $N⊴G$ has prime index $p$, a conjugacy class of $G$ lying in $N$ splits into one or $p$ conjugacy classes of $N$. Conjugation by $G$ acts transitively on these smaller classes, while $N$ fixes every one. This factors through $G/N$, and the orbit-stabilizer theorem makes the number of classes a divisor of $p$.

For a conjugacy class $C$ of a finite group, $z_C={\sum }_{g\in C}g$ is central in its group algebra. In a complex irreducible representation of degree $d$ and character of a representation $\chi$, it acts by $|C|\chi (g)/d$; its trace gives this formula and Schur lemma gives scalarity.

A group $G$ is conjugacy separable when any two nonconjugate elements remain nonconjugate in some finite quotient. Equivalently, every conjugacy class is closed in the profinite topology.

Two group elements $x$ and $y$ are conjugate when $y=gx{g}^{-1}$ for some group element $g$, equivalently when they belong to the same conjugacy class. Conjugate elements have the same order of a group element.

Conjugation by $g$ is the group automorphism $x\mapsto gx{g}^{-1}$. Its orbits are the conjugacy classes.

The group ring $\mathbb{Z}G$ becomes a $\mathbb{Z}G$-module under the conjugation action $g\cdot x=gx{g}^{-1}$. Its basis splits into conjugacy classes, and the span of the class of $x$ is the permutation module on $G/C_G(x)$.

The full symmetry group of a cube acts transitively and faithfully on its twelve edges. For vertices $(\pm 1,\pm 1,\pm 1)$, its signed permutation matrices act transitively on the edges. The stabilizer subgroup of $\{(1,1,z):-1\le z\le 1\}$ consists of independently swapping $x,y$ and reversing $z$, and is a Klein four-group. The orbit-stabilizer theorem gives order $12\cdot 4=48$. Fixing every edge fixes every vertex, as each vertex is the unique intersection of its three incident edges, so the group action is faithful. The resulting subgroup of $S_{12}$ has index $12!/48$ and is not normal: central inversion has cycle type $2^6$, whose conjugacy class in $S_{12}$ has $12!/(2^{6}6!)=10395>48$ elements.

In a finite Galois extension $L/K$, choose a prime ideal $\mathfrak{P}$ above an unramified $\mathfrak{p}$. The Frobenius automorphism at $\mathfrak{P}$ is characterized on its residue field by $x\mapsto x^{N\mathfrak{p}}$. Changing $\mathfrak{P}$ conjugates this element, so its conjugacy class is well-defined. It is the identity class exactly at a completely split prime.

Subgroups $H_1,H_2$ of a finite group $G$ are Gassmann equivalent if $|H_1\cap C|=|H_2\cap C|$ for every conjugacy class $C$. Equivalently their coset permutation representations have the same character of a representation. They need not be conjugate subgroups. The Sunada theorem turns this equality into equality of Laplacian eigenfunction multiplicities.

For a finite group $G$ and representatives $g_i$ of its conjugacy classes, Shapiro's lemma gives

For the dihedral group of a regular $2n$-gon, $n\ge 2$, use $r^{2n}=s^2=1$ and $srs=r^{-1}$. Its involutions are the central half-turn $r^n$ and all reflections $s{r}^j$. The half-turn has singleton conjugacy class and normal closure $\{1,r^n\}$. Conjugating a reflection changes its exponent by an even integer or negates it, so its conjugacy class is $\{s{r}^{j+2k}:0\le k<n\}$. Products of consecutive reflections generate $r^2$, giving normal closure $⟨r^2,s{r}^j⟩$, of order $2n$ and index two. The two reflection classes are distinguished by exponent parity.

For an odd prime $p$, embed the elementary abelian group $(C_p{)}^3$ and the Heisenberg group over a prime field regularly in $S_{p^3}$. Every nonidentity element has order $p$ and hence regular permutation cycle type $p^{p^2}$. The two subgroups have equal intersection counts with every symmetric-group conjugacy class, so they are Gassmann equivalent, but they are not isomorphic because one is abelian and the other is not. Using a compact simply connected isometric cover with ambient deck group $S_{p^3}$ gives quotients with these nonisomorphic fundamental groups; Sunada theorem makes them isospectral while their topology differs.