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This is more or less for a conceptual and better-understanding question in group theory and in representation theory:

(1) Why are conjugacy class and centralizer important concepts in the group / representation theory? What is the important use of conjugacy class and centralizer of an element $g$ in a group $G$?

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(2) How is the case of the conjugacy class and centralizer for an element $g$ in a finite group $G$, different from a continuous group $G'$?

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Note that normal subgroups are unions of conjugacy classes. So conjugacy classes are used to construct normal subgroups. Normal subgroups are used to study groups $G$ by looking at the simpler quotient groups $G/N$, where $N$ needs to be a normal subgroup. –  Mussé Redi Mar 30 at 2:11
    
Sometimes the importance of a concept is revealed if you simply keep on reading. –  Pedro Tamaroff Mar 30 at 2:12
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@MusséRedi If I show you a hammer, you might not find it interesting. If I get you into a carpenter's workshop, maybe you will. –  Pedro Tamaroff Mar 30 at 2:15
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Thanks all for the comments. Is there a similar/analogous concept like Fourier transformation, between conjugacy class and centralizer? (this kind of Fourier transformation works only for finite groups?) –  Idear Mar 30 at 2:17
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@Idear You can use fourier transforms with group characters, which I describe below in my answer. –  Alexander Gruber Mar 30 at 3:08

6 Answers 6

up vote 7 down vote accepted

Abstract groups are fascinating objects on their own, but they are only really useful if you can use them on other structures. To do so, you need the concept of a group action.

Conjugacy classes and centralizers are related to a certain type of group action, a group acting on itself by conjugation - that is, $g\cdot x= g^{-1}xg$. This action gives us a lot of information about the group. The conjugacy classes fit into this picture as the orbits of elements of $G$ under this action, while the centralizers are their stabilizers.

Once you get used to how group actions work, it becomes much easier to see the importance of conjugacy classes and centralizers. For an elementary example, you can tell that $\left|\mathcal{O}_x\right|=[G:C_G(x)]$ for any $x$. It's good to know how many elements a given element is conjugate to, and it's very interesting that this number must divide the order of the group.

For a less elementary example, we can look at how groups act on other sets. A representation of a group $G$ is a homomorphism $G\rightarrow \operatorname{GL}(V)$, where $V$ is a vector space. Usually we are most interested in representations into $\operatorname{GL}_n(\mathbb{C})$. A character is what we call the trace of a representation.

Characters are not necessarily homomorphisms, but, as it turns out, they are class functions, meaning that if $\chi$ is a character, then $\chi(x)=\chi(y)$ whenever $x$ and $y$ are in the same conjugacy class of $G$. So, if we know all the conjugacy classes of the group, we can much more easily compute the characters of the group's representations (and tell a lot of other things about them too). You can see why it would be useful to do this, as symmetries of vector spaces are essential to mathematics in general.

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Thanks. Can you illuminate a bit more on the notation of $$|\mathcal{O}_x|=[G:C_G(x)]$$? :) –  Idear Mar 30 at 15:30
    
It also seems like "Fourier transform on finite groups" works for finite groups. How about continuous groups like SU(2) or SO(3)? Is there a notion of conjugacy class or centralizer there? Thank you! –  Idear Mar 30 at 15:45
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@Idear $[G:C_G(x)]$ is the index of the centralizer of $x$ in $G$. The index of a subgroup is the number of cosets of that subgroup (turns out to be $[G:H]=|G|/|H|$, in finite groups). $|\mathcal{O}_x|$ is the number of elements in the orbit of $x$ under the conjugation action; in other words, the size of the conjugacy class. –  Alexander Gruber Mar 30 at 16:13
    
@Idear Continuous groups have centralizers and conjugacy classes just like any other groups. –  Alexander Gruber Mar 30 at 16:14

I think the conjugacy class is used in proving the Sylow theorems.

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There are several reasons in finite group theory. Conjugacy is a equivalence relation, and therefore breaks up the group as a disjoint union of equivalence classes called conjugacy classes. An important equation is the modified class equation, which states that $|G| = |Z(G)|+ \sum_{i=1}^{n}[G:C_{G}(x_{i})],$ where $x_{1},x_{2}\ldots, x_{n}$ are representatives of the conjugacy classes of size greater than one of $G$ (the conjugacy class of $x$ has size one if and only if $x \in Z(G)).$ This leads quickly to a proof of Sylow's theorem, as Kaa1el implicitly indicated. Another important fact, proved using group characters, is Burnside's Theorem that in a finite non- Abelian simple group no non-identity element can lie in a conjugacy class whose size is a power of a prime. This leads quickly to Burnside's Theorem that a finite group of order $p^{a}q^{b}$ is solvable when $p,q$ are primes, and $a,b$ are positive integers. There are many other uses. In both infinite and finite groups, the notion of conjugacy is useful in exhibiting structure. For example, the decomposition of permutations as products of disjoint cycles makes the relation of conjugacy especially transparent in the symmetric group. The notion of Jordan normal form makes the conjugacy classes in ${\rm GL}(n,F)$ transparent when $F$ is an algebraically closed field. The notion of rational canonical form makes the conjugacy classes in ${\rm GL}(n,F)$ transparent when $F$ is a not necessarily algebraically closed field (Jordan normal form is really a special case of the latter) . There are many, many, other examples- these are just a few illustrations.

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You can prove the simplicity of $A_n$ using the concept of centralizer and conjugacy class, because a normal group is a union of conjugacy classes. Despite being an special case of a more broad concept (i.e. Group Actions), conjugacy have a deep connection with normality.

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Where does the centralizer concept come into play when proving the simplicity of $A_n$? –  Mussé Redi Mar 30 at 2:45
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Rotman`s Introduction to the Theory of Groups uses centralizer and conjugacy classes to prove simplicity of A5, uses simplicity of $A_5$ to prove simplicity of $A_6$ and uses simplicity of $A_6$ to prove simplicity of $A_n$. The key idea is, if $H$ a normal subgroup, it must be the union of conjugacy class, so the size of |H| must be a sum of conjugacy classes sizes, which are given by the index of the centralizer. –  Jonas Gomes Mar 30 at 2:55

A basic attribute to a binary operation is its commutativity i.e. $xax^{-1}=a$ for all $a,x\in G$. When this property does not hold then one asks for the next best, or a measure of failure of commutativity: So if $xax^{-1}$ is considered something closer to $a$ when it is not $a$. They are the elements conjugate to $a$ as $x$ varies in $G$.

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One of the most important concepts in representation theory is induction and restriction. The Frobenius character formula is a useful tool to understand this and has many generalizations to continuous groups (Poisson summation formula, Selberg trace formula, Arthur trace formula). It gives you an identity between irreducible representations with multiplicities and conjugacy classes with volumes of centralizers.

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