---
title: Conjugation and the Class Equation
module: Group Actions and Sylow Theory
moduleNumber: 4
lessonNumber: 2
order: 402
summary: >
  A group acts on itself by conjugation, and the orbits are the conjugacy
  classes. Orbit-stabilizer turns the resulting partition into the class
  equation, which forces every group of prime-power order to have a nontrivial
  center. Conjugacy in the symmetric group is cycle type, and Burnside's lemma
  counts orbits by averaging fixed points.
topics: [Group Actions and Sylow Theory]
sources:
  - book: Dummit & Foote
    ref: "Ch. 4 Group Actions; §4.3 Groups Acting on Themselves by Conjugation — The Class Equation"
  - book: Judson
    ref: "Ch. 14 Group Actions; §14.2 The Class Equation, §14.3 Burnside's Counting Theorem"
draft: false
---

A group acting on itself by conjugation, $g \cdot a = gag^{-1}$, behaves nothing
like the [left-multiplication action](/abstract-algebra/group-actions-and-sylow/actions-and-cayleys-theorem):
it fixes the identity, respects the group operation, and splits $G$ into classes of
unequal size. The sizes obey an arithmetic identity, the **class equation**, which
constrains a group from its order alone.[^df-classeq]

## The conjugation action

Define $G$ acting on itself by $g \cdot a = g a g^{-1}$. The axioms hold:
$$
g_1 \cdot (g_2 \cdot a) = g_1 (g_2 a g_2^{-1}) g_1^{-1} = (g_1 g_2)\, a\, (g_1 g_2)^{-1}
= (g_1 g_2) \cdot a,
\qquad 1 \cdot a = a.
$$

> **Definition (Conjugacy class).** Elements $a, b \in G$ are **conjugate** if
> $b = g a g^{-1}$ for some $g \in G$. The orbits of the conjugation action are the
> **conjugacy classes** of $G$.

Two features distinguish this from left multiplication. First, unless $|G| = 1$
the action is never transitive: $\{1\}$ is always its own class, since $g 1 g^{-1}
= 1$. Second, the singleton classes are precisely the central elements.

> **Observation.** The class of $a$ is $\{a\}$ if and only if $g a g^{-1} = a$ for
> all $g$, that is, if and only if $a \in Z(G)$.

So the center is the union of the size-one orbits. An abelian group is all center:
every class is a singleton, and conjugation is trivial.

The stabilizer of $a$ under conjugation is $\{g : gag^{-1} = a\} = C_G(a)$, the
[centralizer](/abstract-algebra/subgroups-and-quotients/subgroups-and-substructures).
Orbit-stabilizer immediately gives the size of each class.

> **Proposition (Size of a class).** The number of conjugates of $a$ equals
> $[G : C_G(a)]$. More generally, for a subset $S \subseteq G$ acted on by
> conjugation, the number of conjugates of $S$ is $[G : N_G(S)]$, the index of the
> normalizer.

The subset statement uses $N_G(S) = \{g : gSg^{-1} = S\}$ as the stabilizer; for a
single element, $N_G(\{s\}) = C_G(s)$.[^df-prop6] Every class size is therefore a
divisor of $|G|$, a constraint used constantly below.

$$
% caption: $D_8$ splits into five conjugacy classes of sizes $1,1,2,2,2$; the two
% singletons are the center $Z(D_8)=\{1,r^2\}$.
\begin{tikzpicture}[scale=1.0, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\draw[draw=black, dashed] (-0.4,-1.5) rectangle (10.4,1.3);
\node[anchor=south west, font=\footnotesize, text=black] at (-0.35,1.32) {$D_8$};
% center singletons
\node[draw=acc, fill=acc!22, minimum width=10mm, minimum height=8mm] at (0.7,0) {$1$};
\node[draw=acc, fill=acc!22, minimum width=10mm, minimum height=8mm] at (2.2,0) {$r^2$};
\node[font=\footnotesize, text=acc] at (1.45,-1.15) {center};
% size-2 classes
\node[draw=acc, fill=acc!10, minimum width=16mm, minimum height=8mm] at (4.4,0) {$r,\ r^3$};
\node[draw=acc, fill=acc!10, minimum width=18mm, minimum height=8mm] at (6.8,0) {$s,\ sr^2$};
\node[draw=acc, fill=acc!10, minimum width=18mm, minimum height=8mm] at (9.2,0) {$sr,\ sr^3$};
\node[font=\footnotesize, text=black] at (6.8,-1.15) {non-central classes, size $2$};
\end{tikzpicture}
$$

## The class equation

Summing the sizes of all classes recovers $|G|$. Separating the singletons (the
center) from the rest gives the central identity of finite group theory.

> **Theorem (Class equation).** Let $G$ be finite and let $g_1, \dots, g_r$
> represent the distinct conjugacy classes _not_ contained in $Z(G)$. Then
>
> $$
> |G| = |Z(G)| + \sum_{i=1}^{r} [G : C_G(g_i)].
> $$

Each central element is its own class, contributing $|Z(G)|$; each non-central
class $g_i$ contributes $[G : C_G(g_i)] > 1$. The classes partition $G$, so the
sizes sum to $|G|$.[^df-thm7] Every summand on the right divides $|G|$, and every
non-central summand exceeds $1$; together these two constraints are strongly
restrictive.

$$
% caption: The class equation as a length: $|G|$ decomposes into the center (unit
% blocks) plus non-central class sizes, each a divisor of $|G|$ exceeding one.
\begin{tikzpicture}[scale=1.0, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
% center unit blocks
\foreach \i in {0,1} \draw[fill=acc!22, draw=acc] (\i*0.7,0) rectangle (\i*0.7+0.7,0.8);
\node[font=\footnotesize, text=acc, anchor=north] at (0.7,-0.15) {center $Z(G)$};
% class blocks
\draw[fill=acc!10, draw=acc] (1.4,0) rectangle (2.8,0.8);
\node at (2.1,0.4) {$[G{:}C_G(g_1)]$};
\draw[fill=acc!10, draw=acc] (2.8,0) rectangle (4.6,0.8);
\node at (3.7,0.4) {$[G{:}C_G(g_2)]$};
\draw[fill=acc!10, draw=acc] (4.6,0) rectangle (6.7,0.8);
\node at (5.65,0.4) {$[G{:}C_G(g_3)]$};
\node[font=\footnotesize, text=black, anchor=north] at (4.1,-0.15) {non-central classes};
% total span line with end ticks
\draw[black] (0,1.05) -- (6.7,1.05);
\draw[black] (0,0.92) -- (0,1.18);
\draw[black] (6.7,0.92) -- (6.7,1.18);
\node[font=\footnotesize, anchor=south] at (3.35,1.2) {total: order of $G$};
\end{tikzpicture}
$$

For the two nonabelian groups of order $8$, a shortcut speeds the computation:
$\langle g \rangle \le C_G(g)$ always, so a non-central element of $D_8$ or $Q_8$
has centralizer of order exactly $4$ and hence a class of size $2$.

- **$Q_8$.** The classes are $\{1\}, \{-1\}, \{\pm i\}, \{\pm j\}, \{\pm k\}$, with
  $Z(Q_8) = \{1, -1\}$. The class equation is $8 = 2 + 2 + 2 + 2$.
- **$D_8$.** The classes are $\{1\}, \{r^2\}, \{r, r^3\}, \{s, sr^2\}, \{sr, sr^3\}$,
  with $Z(D_8) = \{1, r^2\}$. The class equation is again $8 = 2 + 2 + 2 + 2$.

Both groups share the same class-size profile even though they are not isomorphic:
the class equation constrains a group without determining it.

### Prime-power order forces a center

The first consequence underlies the theory of
[$p$-groups](/abstract-algebra/products-and-group-structure/nilpotent-and-solvable-groups).

> **Theorem ($p$-groups have nontrivial center).** If $|P| = p^a$ for a prime $p$
> and $a \ge 1$, then $Z(P) \ne 1$.

Read the class equation modulo $p$. Each non-central summand $[P : C_P(g_i)]$ is a
divisor of $p^a$ greater than $1$, hence divisible by $p$. Since $p \mid |P|$ as
well, the equation forces $p \mid |Z(P)|$. In particular $|Z(P)| \ge p$, so the
center is nontrivial.[^df-thm8] A group with $Z(P) = 1$ cannot have prime-power
order.

> **Corollary (Groups of order $p^2$).** If $|P| = p^2$, then $P$ is abelian, and
> $P \cong \mathbb{Z}_{p^2}$ or $P \cong \mathbb{Z}_p \times \mathbb{Z}_p$.

Since $Z(P) \ne 1$ and $|Z(P)|$ divides $p^2$, if $Z(P) \ne P$ then $P / Z(P)$ has
order $p$, hence is cyclic — but a group with cyclic central quotient is abelian, a
contradiction. So $P = Z(P)$ is abelian, and the classification of finite abelian
groups splits it into the two listed types.[^df-cor9]

> **Worked example.** Read the center of $Q_8$ off its class equation. Each
> non-central element commutes with its own powers, so $C_{Q_8}(i) = \{1, -1, i, -i\}$
> has order $4$, and the class of $i$ has size $[Q_8 : C_{Q_8}(i)] = 2$; the same
> holds for $j$ and $k$. Collecting the classes,
>
> $$
> 8 = \underbrace{2}_{|Z(Q_8)|} + 2 + 2 + 2,
> \qquad Z(Q_8) = \{1, -1\}.
> $$
>
> Reduced modulo $2$, every non-central summand is even, forcing $2 \mid |Z(Q_8)|$
> — the prime-power center theorem specialized to $|Q_8| = 2^3$.[^df-thm8]

## Conjugacy in the symmetric group

Conjugation in $S_n$ has an explicit description: it relabels the entries of a
cycle decomposition.[^df-prop10]

> **Proposition (Conjugation relabels).** If $\sigma \in S_n$ has cycle
> decomposition $(a_1\, a_2\, \cdots)(b_1\, b_2\, \cdots)\cdots$ and $\tau \in
> S_n$, then $\tau \sigma \tau^{-1}$ has cycle decomposition
> $(\tau(a_1)\, \tau(a_2)\, \cdots)(\tau(b_1)\, \cdots)\cdots$ — each entry $i$
> replaced by $\tau(i)$.

The proof is one line: if $\sigma(i) = j$ then $\tau\sigma\tau^{-1}(\tau(i)) =
\tau(j)$, so the ordered pair $\tau(i), \tau(j)$ appears in the conjugate exactly
where $i, j$ appeared in $\sigma$. The **cycle type** — the multiset of cycle
lengths (including fixed points as $1$-cycles) — is a complete invariant.

> **Proposition (Cycle type classifies).** Two permutations of $S_n$ are conjugate
> if and only if they have the same cycle type. The number of conjugacy classes of
> $S_n$ equals the number of partitions of $n$.

Same cycle type gives a matching between the two decompositions; the permutation
$\tau$ carrying one list to the other conjugates one into the other. A cycle type
is precisely a [partition](/abstract-algebra/foundations/integers-and-modular-arithmetic)
of $n$, so the classes are indexed by partitions.[^df-prop11]

$$
% caption: The five partitions of $4$, drawn as columns, index the five conjugacy
% classes of $S_4$; each column height list is a cycle type.
\begin{tikzpicture}[scale=0.62, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% partition 1+1+1+1 : four columns of height 1
\foreach \x in {0,1,2,3} \fill[acc] (\x,0) circle (3.4pt);
\node[text=black] at (1.5,-1.1) {$1{+}1{+}1{+}1$};
% partition 2+1+1
\begin{scope}[xshift=5.4cm]
\fill[acc] (0,0) circle (3.4pt); \fill[acc] (0,1) circle (3.4pt);
\fill[acc] (1,0) circle (3.4pt); \fill[acc] (2,0) circle (3.4pt);
\node[text=black] at (1,-1.1) {$2{+}1{+}1$};
\end{scope}
% partition 2+2
\begin{scope}[xshift=10.2cm]
\fill[acc] (0,0) circle (3.4pt); \fill[acc] (0,1) circle (3.4pt);
\fill[acc] (1,0) circle (3.4pt); \fill[acc] (1,1) circle (3.4pt);
\node[text=black] at (0.5,-1.1) {$2{+}2$};
\end{scope}
% partition 3+1
\begin{scope}[xshift=13.6cm]
\fill[acc] (0,0) circle (3.4pt); \fill[acc] (0,1) circle (3.4pt); \fill[acc] (0,2) circle (3.4pt);
\fill[acc] (1,0) circle (3.4pt);
\node[text=black] at (0.5,-1.1) {$3{+}1$};
\end{scope}
% partition 4
\begin{scope}[xshift=16.8cm]
\foreach \y in {0,1,2,3} \fill[acc] (0,\y) circle (3.4pt);
\node[text=black] at (0,-1.1) {$4$};
\end{scope}
\end{tikzpicture}
$$

The class sizes come from the centralizer formula. For an $m$-cycle $\sigma$ in
$S_n$, the number of $m$-cycles is $\frac{n(n-1)\cdots(n-m+1)}{m}$, so
$|C_{S_n}(\sigma)| = m \cdot (n-m)!$: the $m$-cycle commutes with its own powers
and with any permutation disjoint from it. Working this out for $S_4$:

| Cycle type | Representative | Class size | Order | Even? |
| --- | --- | --- | --- | --- |
| $1{+}1{+}1{+}1$ | identity | $1$ | $1$ | yes |
| $2{+}1{+}1$ | $(1\,2)$ | $6$ | $2$ | no |
| $2{+}2$ | $(1\,2)(3\,4)$ | $3$ | $2$ | yes |
| $3{+}1$ | $(1\,2\,3)$ | $8$ | $3$ | yes |
| $4$ | $(1\,2\,3\,4)$ | $6$ | $4$ | no |

The sizes sum to $1 + 6 + 3 + 8 + 6 = 24 = |S_4|$, and $Z(S_4) = 1$ (the only
singleton class is the identity). The even classes — sizes $1, 3, 8$ — sum to
$12 = |A_4|$, which is why normal subgroups, being unions of classes, are so
constrained; this counting drives the
[simplicity arguments](/abstract-algebra/group-actions-and-sylow/automorphisms-and-simple-groups)
for the alternating groups.

One subtlety carries into the alternating group: a class of $S_n$ contained in
$A_n$ need not remain a single class under conjugation _by $A_n$ alone_, since the
permutation $\tau$ relabeling one representative to another may be odd. The class
splits in two exactly when its cycle type consists of distinct odd integers. For
$S_5$ the classes have sizes $1, 10, 15, 20, 20, 30, 24$ (partitions $1^5, 2\,1^3,
2^2 1, 3\,1^2, 3\,2, 4\,1, 5$), and inside $A_5$ the even classes are $1, 15, 20$ and the
$24$ five-cycles, which split into two classes of $12$. Those sizes $1, 15, 20, 12,
12$ are what make $A_5$ simple: no sub-collection containing the $1$ sums to a
proper divisor of $60$.

## Burnside's lemma: counting orbits

Orbit-stabilizer counts one orbit; Burnside's lemma counts _all_ of them at once,
by averaging fixed points. For an action of $G$ on $X$, let $X^g = \{x \in X : g
\cdot x = x\}$ be the set fixed by $g$.

> **Theorem (Burnside / orbit-counting).** If a finite group $G$ acts on a finite
> set $X$, the number $k$ of orbits is
>
> $$
> k = \frac{1}{|G|} \sum_{g \in G} |X^g|.
> $$

Count the pairs $(g, x)$ with $g \cdot x = x$ two ways. Summing over $g$ gives
$\sum_g |X^g|$; summing over $x$ gives $\sum_x |G_x|$. For each orbit
$\mathcal{O}$, orbit-stabilizer makes $\sum_{x \in \mathcal{O}} |G_x| = |\mathcal{O}|
\cdot |G_x| = |G|$, so summing over the $k$ orbits gives $k \cdot |G|$.[^judson-burnside]
Equating the two counts and dividing by $|G|$ finishes it.

The application is counting configurations "up to symmetry." Consider a necklace of
four beads, each coloured black or white, where two colourings are the same if one
rotates to the other. The group is the cyclic rotation group $C_4 = \{e, \rho,
\rho^2, \rho^3\}$ acting on the $2^4 = 16$ colourings, and a colouring is fixed by
$\rho^d$ exactly when it is constant on the orbits of $\rho^d$ on the beads.

$$
% caption: A four-bead necklace under rotation by $90^\circ$: two of the sixteen
% two-colour patterns that Burnside's average collapses into six distinct necklaces.
\begin{tikzpicture}[scale=1.0, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
% necklace 1
\foreach \a/\fill in {90/acc, 0/white, 270/white, 180/white} {
  \fill[\fill, draw=black] (\a:0.9) circle (0.24);
}
\draw[black] (0,0) circle (0.9);
\draw[->, acc, thick] (1.35,0.4) arc (25:-25:0.9);
% necklace 2 (rotated)
\begin{scope}[xshift=3.4cm]
\foreach \a/\fill in {90/white, 0/acc, 270/white, 180/white} {
  \fill[\fill, draw=black] (\a:0.9) circle (0.24);
}
\draw[black] (0,0) circle (0.9);
\end{scope}
\node[font=\footnotesize, text=black, anchor=west] at (5.0,0) {same necklace, up to a quarter turn};
\end{tikzpicture}
$$

The per-element fixed-colouring counts follow the cycle structure of each rotation
acting on the four bead positions:

| Rotation | Cycles on beads | Fixed colourings $\lvert X^g\rvert$ |
| --- | --- | --- |
| $e$ | $(1)(2)(3)(4)$ | $2^4 = 16$ |
| $\rho$ | $(1\,2\,3\,4)$ | $2^1 = 2$ |
| $\rho^2$ | $(1\,3)(2\,4)$ | $2^2 = 4$ |
| $\rho^3$ | $(1\,4\,3\,2)$ | $2^1 = 2$ |

Averaging, $k = \tfrac{1}{4}(16 + 2 + 4 + 2) = 6$: there are exactly six distinct
two-colour necklaces. A fixed colouring under a rotation whose cycle decomposition
has $c$ cycles must be constant on each cycle, giving $|X^g| = 2^{c}$ colours —
which is why the fixed counts are powers of two indexed by the number of cycles.
The same average recurs whenever configurations must be counted up to symmetry.

> **Worked example.** Count the two-colourings of the six faces of a cube, up to
> rotation. The [rotation group of the cube](/abstract-algebra/group-actions-and-sylow/actions-and-cayleys-theorem)
> has order $24$, and Burnside averages the fixed colourings over its five rotation
> types, sorted by how each permutes the six faces:
> - identity ($1$ element): all $2^6 = 64$ colourings fixed;
> - face turns by $90^\circ$ or $270^\circ$ ($6$): a $4$-cycle on the side faces and
>   $2$ fixed faces give $3$ cycles, so $2^3 = 8$ each;
> - face turns by $180^\circ$ ($3$): two $2$-cycles and $2$ fixed faces give $4$
>   cycles, so $2^4 = 16$ each;
> - vertex turns by $120^\circ$ or $240^\circ$ ($8$): two $3$-cycles on the faces
>   give $2$ cycles, so $2^2 = 4$ each;
> - edge turns by $180^\circ$ ($6$): three $2$-cycles give $3$ cycles, so $2^3 = 8$
>   each.
>
> $$
> k = \frac{1}{24}\bigl(64 + 6\cdot 8 + 3\cdot 16 + 8\cdot 4 + 6\cdot 8\bigr)
> = \frac{240}{24} = 10.
> $$
>
> There are ten distinct face-colourings.[^judson-burnside]

[^df-classeq]: **Dummit & Foote**, §4.3 — Groups Acting on Themselves by Conjugation: the conjugation action, conjugacy classes, and the center as the union of singleton classes.
[^df-prop6]: **Dummit & Foote**, §4.3, Proposition 6: the number of conjugates of a subset (element) is the index of its normalizer (centralizer).
[^df-thm7]: **Dummit & Foote**, §4.3, Theorem 7: the class equation.
[^df-thm8]: **Dummit & Foote**, §4.3, Theorem 8: a group of prime-power order has a nontrivial center.
[^df-cor9]: **Dummit & Foote**, §4.3, Corollary 9: groups of order $p^2$ are abelian, isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_p \times \mathbb{Z}_p$.
[^df-prop10]: **Dummit & Foote**, §4.3, Proposition 10: conjugation in $S_n$ relabels the entries of the cycle decomposition.
[^df-prop11]: **Dummit & Foote**, §4.3, Proposition 11: two elements of $S_n$ are conjugate iff they share a cycle type; classes correspond to partitions of $n$.
[^judson-burnside]: **Judson**, §14.3, Theorem 14.7 — Burnside's Counting Theorem: the number of orbits equals the average number of fixed points, with the square/necklace colouring application.