Conjugacy

Two elements $x,y$ of a group $G$ are conjugate if there exists $g\in G$ with

$$y=g^{-1}xg.$$

Equivalently, $x,y$ lie in the same orbit of the conjugation action $\lambda (g)x=gx{g}^{-1}$ of $G$ on itself. The orbit of $x$ under conjugation is its conjugacy class:

$$\mathrm{Cl}(x)=\{gx{g}^{-1}:g\in G\}.$$

Centraliser

The Stabiliser of $x$ under conjugation is the centraliser

$$C_G(x)=\{g\in G:gx=xg\},$$

the elements of $G$ that commute with $x$.

By the Orbit-Stabiliser theorem:

$$|\mathrm{Cl}(x)|=|G|/|C_G(x)|.$$

Properties

• Conjugacy is an equivalence relation on $G$ - its equivalence classes partition $G$.
• The class of the identity is $\{e\}$.
• More generally, $x$ has a singleton class iff $x$ commutes with everything, i.e. $x\in Z(G)$ (the centre).
• Conjugate elements have the same order, the same cycle shape (in $S_n$), and the same characteristic polynomial (in $\mathrm{GL}$).

Class Equation

Partitioning $G$ into conjugacy classes:

$$|G|=|Z(G)|+\sum _i\frac{|G|}{|C_G(g_i)|},$$

summed over representatives $g_i$ of the non-central conjugacy classes. This is the class equation - the workhorse of much classification of finite groups (e.g. proving non-trivial centre for $p$-groups).

In $S_n$: Cycle Shape

In the symmetric group, two permutations are conjugate iff they have the same cycle shape. For example:

• $(123)$ and $(456)$ are conjugate in $S_6$ (both 3-cycles, with the others fixed).
• $(12)(34)$ and $(1234)$ are not conjugate (different shapes).

Concretely: if $\pi =(a_1{a}_2\cdots a_r)\cdots$ in $S_n$ and $g\in S_n$, then $g\pi g^{-1}=(g(a_1)g(a_2)\cdots g(a_r))\cdots$ - same shape, vertices relabelled.

Normaliser

For a subset $S\subseteq G$, the normaliser is

$$N_G(S)=\{g\in G:gS{g}^{-1}=S\}.$$

For a subgroup $H\le G$, $N_G(H)$ is the largest subgroup of $G$ in which $H$ is normal.

Use in Sylow Theory

The proofs of Sylow’s theorems act $G$ on the set of all Sylow $p$-subgroups by conjugation, and use the orbit-stabiliser correspondence with normalisers and centralisers.