Group theory MOC

Conjugation by an element

In a group 𝐺, given elements 𝑥,𝑔 ∈𝐺, we may conjugate 𝑥 by 𝑔 to get 𝑔𝑎𝑔−1. Sometimes this is written as ˆ𝑏𝑎 or 𝑏𝑎, or for the right-action variant, 𝑎𝑏 =𝑏−1𝑎𝑏.

Conjugation as an action

Conjugation by a given element is an automorphism of the group, such that ̂⋅ :𝐺 →Aut⁡(𝐺) constitutes a group action. The orbit of an element 𝑥 ∈𝐺 is its Conjugacy class [𝑥]∼, while its Stabilizer group is its centralizer group. An automorphism given by conjugation is called an inner automorphism, and the image ̂𝐺 =Inn(𝐺) ⊴Aut⁡(𝐺) forms the inner automorphism group.

Conjugacy relation

Given two group elements 𝑥,𝑦 ∈𝐺, we say 𝑥 is conjugate to 𝑦 (𝑥 ∼𝑦) iff there exists 𝑔 ∈𝐺 such that 𝑦 =𝑔𝑥𝑔−1. group The conjugacy relation ∼ is an Equivalence relation. group

Proof of equivalence relation

For any 𝑥 ∈𝐺, 𝑥 =𝑒𝑥𝑒−1 ⟹ 𝑥 ∼𝑥, thus ∼ is reflexive. For any 𝑥,𝑦 ∈𝐺, 𝑥 ∼𝑦 ⟺ 𝑦 =𝑔𝑥𝑔−1 ⟺ 𝑥 =(𝑔−1)𝑦(𝑔−1)−1 ⟺ 𝑦 ∼𝑥, thus ∼ is symmetric. For any 𝑥,𝑦,𝑧 ∈𝐺 such that 𝑥 ∼𝑦 and 𝑦 ∼𝑧, there exist 𝑔,ℎ ∈𝐺 such that 𝑦 =𝑔𝑥𝑔−1 and 𝑧 =ℎ𝑦ℎ−1. Then 𝑧 =ℎ𝑔𝑥𝑔−1ℎ−1 =(ℎ𝑔)𝑥(ℎ𝑔)−1 and hence 𝑥 ∼𝑧, wherefore ∼ is transitive. Therefore ∼ is an equivalence relation.

A conjugacy relation may also be applied between subgroups, see Conjugate subgroups.

Conjugacy class

The equivalence classes for the conjugacy relation form so-called conjugacy classes.

[𝑥]∼={𝑔𝑥𝑔−1:𝑔∈𝐺}

Properties

See also Inner group automorphism.

• [𝑒]∼ ={𝑒} since 𝑔𝑒𝑔1 =𝑒 for all 𝑔 ∈𝐺.
• ˆ𝑔𝑥 =𝑥 iff 𝑔 and 𝑥 commute
• From above it follows that in an Abelian group all conjugacy classes are singletons.
• A conjugacy class is not necessarily a subgroup (since it is either the trivial subgroup {𝑒} or 𝑒 ∉[𝑥]∼).
• By the Orbit-stabilizer theorem, |𝐶(𝑥)| ⋅|[𝑥]∼| =|𝐺|.
• The number of conjugacy classes equals the number of non-equivalent irreps of a group.

Examples

• In SO(3) rotations by the same angle (i.e. only differing in axis of rotation) form conjugacy classes.
• In GL(𝑛) elements are conjugate to each other iff they have similar matrices (in subgroups, such as SO(𝑛), conjugacy may be more restricted, however all conjugate elements are similar).

---

tidy| en | SemBr