Conjugacy begins with a nuisance that turns into a method: the same group element can look different after the group renames itself from the inside. In a symmetry group, this is not a cosmetic relabelling imposed from outside. It is the operation of taking one symmetry, moving the whole system by another symmetry, performing the first symmetry, and moving back.

In $S_n$, this internal relabelling preserves the cycle pattern of a permutation. In matrix groups, it is the algebraic version of changing basis. In every group, it divides elements into families whose members have the same internal type. Conjugacy classes are those families.

[example: Relabelling a Transposition in $S_4$] Let $G = S_4$, let $g = (1\,2)$, and let $h = (1\,3\,4)$. Then $h^{-1} = (1\,4\,3)$, and since permutations act on the left, the composite $hgh^{-1}$ is evaluated from right to left. We compute its value on each symbol: \begin{align*} (hgh^{-1})(1) &= h(g(h^{-1}(1))) = h(g(4)) = h(4) = 1, \\ (hgh^{-1})(2) &= h(g(h^{-1}(2))) = h(g(2)) = h(1) = 3, \\ (hgh^{-1})(3) &= h(g(h^{-1}(3))) = h(g(1)) = h(2) = 2, \\ (hgh^{-1})(4) &= h(g(h^{-1}(4))) = h(g(3)) = h(3) = 4. \end{align*} Thus $hgh^{-1}$ fixes $1$ and $4$, sends $2$ to $3$, and sends $3$ to $2$, so \begin{align*} h(1\,2)h^{-1} = (2\,3). \end{align*} Equivalently, conjugation has transported the two moved letters $1$ and $2$ to $h(1)=3$ and $h(2)=2$. The result is still a transposition, showing in this case that conjugacy in $S_n$ preserves the shape of a permutation while renaming its symbols. [/example]

This relabelling principle is not limited to permutations. It is the basic way a group acts on itself while respecting its multiplication. The chapter develops conjugacy classes as orbits, counts them by stabilisers, computes them in important groups, and explains why they control normal subgroups and character theory.

## Definition

The first object to isolate is the motion itself. If $h$ is allowed to change the internal viewpoint, then $g$ is transported to $hgh^{-1}$. This operation is designed so that multiplication relations are preserved while the element being studied is moved to a different representative of the same internal type.

Let $G$ be a group. Elements $g,k \in G$ are called conjugate in $G$ if there exists $h \in G$ such that $k = hgh^{-1}$. The ambient group is part of the data: two elements may become conjugate after enlarging the group, because the larger group supplies more elements $h$ with which to perform the transport.

Once a single transported element is understood, the natural question is how far a fixed element can move. The answer is not a number at first, but a subset of the group: the whole family of elements obtained by internal relabelling.

[definition: Conjugacy Class] Let $G$ be a group and let $g \in G$. The conjugacy class of $g$ in $G$ is the subset \begin{align*} \operatorname{Cl}_G(g) = \{hgh^{-1} : h \in G\} \subset G. \end{align*} [/definition]

The definition gives the object, but it does not yet tell us whether conjugacy really partitions the group into coherent pieces. The next theme supplies that structure.

## Conjugacy as an Equivalence Relation

A classification by internal type should have three basic features: every element should have its own type, the order of comparison should not matter, and two comparisons through an intermediate representative should compose. These are exactly the axioms of an [equivalence relation](/page/Equivalence%20Relation).

[quotetheorem:5005]

This theorem says that the conjugacy classes are not overlapping fragments. They form a partition of $G$, so every element belongs to exactly one conjugacy class.

[example: The Conjugacy Classes of $S_3$] Let $G=S_3$. For every $h \in S_3$, \begin{align*} heh^{-1}(x) &= h(e(h^{-1}(x))) \\ &= h(h^{-1}(x)) \\ &= x \end{align*} for each $x \in \{1,2,3\}$, so $heh^{-1}=e$ and $\operatorname{Cl}_{S_3}(e)=\{e\}$. For a transposition $(a\,b)$ and any $h \in S_3$, the conjugate $h(a\,b)h^{-1}$ sends $h(a)$ to $h(b)$, sends $h(b)$ to $h(a)$, and fixes the remaining symbol. Hence \begin{align*} h(a\,b)h^{-1}=(h(a)\,h(b)). \end{align*} Applying this to $(1\,2)$ gives \begin{align*} e(1\,2)e^{-1} &= (1\,2), \\ (1\,3)(1\,2)(1\,3)^{-1} &= ((1\,3)(1)\,(1\,3)(2)) = (3\,2) = (2\,3), \\ (2\,3)(1\,2)(2\,3)^{-1} &= ((2\,3)(1)\,(2\,3)(2)) = (1\,3). \end{align*} Thus the conjugacy class of $(1\,2)$ is \begin{align*} \{(1\,2),(1\,3),(2\,3)\}. \end{align*} For the $3$-cycle $(1\,2\,3)$, the same relabelling rule gives \begin{align*} h(1\,2\,3)h^{-1}=(h(1)\,h(2)\,h(3)). \end{align*} Taking $h=e$ gives $(1\,2\,3)$, while taking $h=(1\,2)$ gives \begin{align*} (1\,2)(1\,2\,3)(1\,2)^{-1} &= ((1\,2)(1)\,(1\,2)(2)\,(1\,2)(3)) \\ &= (2\,1\,3) \\ &= (1\,3\,2). \end{align*} These are the only two $3$-cycles in $S_3$, so their class is \begin{align*} \{(1\,2\,3),(1\,3\,2)\}. \end{align*} Therefore $S_3$ is partitioned into conjugacy classes of sizes $1$, $3$, and $2$. [/example]

The partition of $S_3$ already hints that conjugacy classes measure non-commutativity. The identity behaves like a central element, while transpositions and $3$-cycles move in families under relabelling.

The equivalence relation viewpoint tells us that conjugacy classes partition $G$, but it does not explain their sizes. For counting, the orbit viewpoint is stronger.

## Orbits, Stabilisers, and the Class Equation

### Centralisers as Stabilisers

Before counting the elements that fail to move a chosen $g$, it is useful to isolate the elements that are never moved by conjugation. These are precisely the elements that commute with everything in the group, and they form the fixed part of the conjugation action.