Let a group act on itself by conjugation, $g \cdot x = gxg^{-1}$. The orbits are called conjugacy classes, and counting them gives an equation with real consequences for groups of prime-power order.
Definitions. The conjugacy class of $x$ is $\{gxg^{-1} : g \in G\}$. The stabilizer of $x$ is its centralizer $C(x) = \{g : gx = xg\}$. The class of $x$ is $\{x\}$ alone exactly when $x$ commutes with everything, that is, when $x$ is in the center $Z(G)$.
Why it works
Application. If $|G| = p^{n}$ with $p$ prime and $n \ge 1$, then $Z(G) \ne \{e\}$. Proof: each $[G : C(x_{i})]$ is a divisor of $p^{n}$ bigger than $1$, so a multiple of $p$. Then $|Z(G)| = |G| - \sum [G : C(x_{i})]$ is a multiple of $p$, and it is at least $1$ (it contains $e$), so it is at least $p$. $\square$
Classes in $S_{n}$. Conjugating relabels: $\sigma(a_{1}\;\dots\;a_{k})\sigma^{-1} = (\sigma(a_{1})\;\dots\;\sigma(a_{k}))$. So two permutations are conjugate exactly when they have the same cycle type, and the classes of $S_{n}$ match the partitions of $n$.
| Group | Class equation |
|---|---|
| $S_{3}$ | $6 = 1 + 2 + 3$ |
| $D_{4}$ | $8 = 1 + 1 + 2 + 2 + 2$ |
| $Q_{8}$ | $8 = 1 + 1 + 2 + 2 + 2$ |
| $S_{4}$ | $24 = 1 + 3 + 6 + 6 + 8$ |
| $A_{4}$ | $12 = 1 + 3 + 4 + 4$ |
Worked example: The class equation of $D_{4}$
- $Z(D_{4}) = \{e, r^{2}\}$: the half turn commutes with everything.
- $srs^{-1} = r^{-1} = r^{3}$, so $\{r, r^{3}\}$ is a class.
- $rsr^{-1} = r^{2}s$, so $\{s, r^{2}s\}$ is a class, and likewise $\{rs, r^{3}s\}$.
- $8 = 1 + 1 + 2 + 2 + 2$.
Answer: Five classes.
Worked example: A class in $S_{4}$
- All permutations of the same cycle type: two disjoint transpositions.
- Pair up $\{1, 2, 3, 4\}$: $(1\;2)(3\;4)$, $(1\;3)(2\;4)$, $(1\;4)(2\;3)$.
Answer: $3$.
Common mistake
Key idea