Conjugation action

A group acts on itself and on its subgroups by conjugation: $g\cdot x=gx{g}^{-1}$ and $g\cdot H=gH{g}^{-1}$.

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

For a subset $S$ of a group and an element $g$, its conjugate subset isConjugation is a bijection and a group automorphism, so it preserves cardinality and every property expressed using the group operation.

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.

The conjugacy class of $x\in G$ is its orbit under the conjugation action:

If an infinite group has a single nonidentity conjugacy class, it is a torsion-free group. Otherwise every nonidentity element has the same finite order $n$. Powers show $n$ is prime. For $n\ge 3$, choose $xg{x}^{-1}=g^2$; then $x^n=1$ gives $g=g^{2^n}$, impossible by Fermat's little theorem. For $n=2$, the group has exponent two and is abelian, so every conjugacy class is a singleton and the hypothesis would force the group to have only two elements. Infinitude excludes this final case.

A shortest conjugacy representative is a word having the fewest written letters among all words representing elements of a fixed conjugacy class. This minimum equals the least group word length of an element in the class; a longer spelling of that element is not itself a shortest representative. Such a word exists because lengths are nonnegative integers. It is freely and cyclically reduced: free cancellation shortens the same representative, while removing mutually inverse first and last letters shortens a conjugate. Every cyclic rotation has the same minimal length.

A nonempty shortest conjugacy representative is a cyclically reduced word. If $w=sv{s}^{-1}$ with first and last letters inverse, $v$ is a shorter word representing a conjugate. Consequently $w^n$ has length exactly $n|w|$ as a freely reduced word, and any segment of the periodic word of length at most $|w|$ fits in a cyclic rotation of $w$. This observation turns a Dehn shortening segment crossing copy boundaries into a shortening of a conjugacy representative.

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 an even permutation $g$, its symmetric group class splits into two equal alternating group classes exactly when its symmetric-group centralizer contains no odd permutation. If an odd commuting element exists, an odd conjugator can be multiplied by it to become even; otherwise the index-two subgroup has two conjugation orbits. In cycle notation the splitting types are precisely distinct odd cycle lengths, counting fixed points as length-one cycles: rotating odd cycles is even, whereas an even cycle or the exchange of two equal odd cycles supplies an odd centralizer element.

For a finite group,where the $x_i$ represent the noncentral conjugacy classes.

For a finite group with trivial centre of a group and a prime divisor $p$ of its order, some nonidentity conjugacy class has size coprime to $p$. Otherwise the class equation would give $|G|\equiv 1(\mathrm{mod}p)$. Primality is essential: in the symmetric group $S_3$, neither nonidentity class size is coprime to the composite divisor $6$.

The centralizer of $x\in G$ is the stabilizer of $x$ under the conjugation action:

Commutation with a fixed-point-free involution preserves its partner relation. A commuting permutation can permute the $m$ pairs and independently swap each pair. This gives the permutation wreath product with order $2^{m}m!$.

A permutation commutes with a transposition exactly when it preserves its two-element support as a set. It may swap those two points and independently permute the remaining points. Thus this centraliser has order $2(n-2)!$.

The normalizer $N_G(H)=\{g\in G:gH{g}^{-1}=H\}$ is the stabilizer of a subgroup $H$ under the conjugation action.

Over the reals or a finite field with more than two elements, the normalizer of all invertible diagonal matrices in $G{L}_2$ consists of monomial matrices and has quotient $C_2$ over that diagonal subgroup. Over ${\mathbb{F}}_p$ its size is $2(p-1{)}^2$ for $p>2$. For $p=2$ the diagonal subgroup is trivial, so its normalizer is all of $G{L}_2({\mathbb{F}}_2)$, of order six.