Is there a systematic way of finding the conjugacy class and/or centralizer of an element?

Is there a systematic way of finding the conjugacy class and centralizer of an element? Could the task be simplified if we are working with "special groups" such as $S_n$ or $A_n$? Are there any intuitive approaches? Thanks.

Well, I am happy you call the groups, $S_n$ and $A_n$ special. They are!

So, I'll describe a method for calculating the conjugacy class and hence the centralizer in $S_n$.

Step 1:

This comes from realising that any two elements of the same cycle type are conjugate to each other in $S_n$. How do we prove this startling result?

Proof.

Lemma

For any $\tau, \sigma \in S_n$, since, $\sigma$ is a product of disjoint cycles, let's say for instance, that, cycle decomposition of $\sigma $ is given by $$\sigma=(a_1a_2a_3\cdots a_k)(b_1b_2b_3\cdots b_l)\cdots$$

We claim that $\tau\sigma\tau^{-1}=(\tau(a_1)\tau(a_2)\cdots\tau(a_k))(\tau(b_1)\tau(b_2)\cdots\tau(b_l))\cdots$

Proof of Lemma: The proof is subtle. Suppose $i \overset{\sigma}{\mapsto}j$, we'll prove that $\tau(i) \overset{\tau\sigma\tau^{-1}}{\mapsto}\tau(j)$.

Now $$\begin{align*}\tau\sigma\tau^{-1}(\tau(i))&=\tau(\sigma(i))\\&=\tau(j)\end{align*}$$ This proves the claim. $\diamond$

So, the cycle type of an element and its conjugate are the same. So, this proves that all elements of the same cycle type are conjugate to each other.$~~~~~~~~~~\blacksquare$

Step-2

This step is a little intuitive. We claim that the number of conjugacy classes in $S_n$ equals the partition of $n$.

Proof.

Firstly, let's prove that the number of cycle types in $S_n$ equal the number of partitions of $n$. The proof of this is rather intuitive. With any permutation, associate the partition whose parts equal the number of elements in a cycle. This gives you the required bijection between the cycle types of $S_n$ and partition of $n$.

To get a feel for it, look at the following in $S_4$:

$$\begin{align*}(1234)&\cong 4\\(12)(34) &\cong 2+2\\ (34) &\cong 1+1+2(\text{since (1) and (2) are omitted in this notation})\end{align*}$$

So, now, since two elements of the same cycle type are conjugate in $S_n$$^\dagger$, we have that they belong to the same conjugacy class. So, counting the number of cycle types gives you the number of conjugacy classes. So, we have that the number of conjugacy classes is equal to the number of partitions of $n$. $\blacksquare$

So, we have so far described the conjugacy classes in $S_n$. But, unfortunately, human brain cannot defeat the symmetry in nature, thanks to people like Ramanujan, little atleast is known about $p(n)$, the number of partitions of $n$. Among the little, a closed form formula is not one!

Now, use the orbit-stabilizer lemma, with the understanding that stabilizer of a point under conjugacy is nothing but its centralizer.

For a specific example such as the one you asked Gerry through comments, you can actually work through with this description.

For instance, $(1234567)$ in $S_n$

A little more machinery is involved! You need to find the number of $r$-cycles in $S_n$ for appropriate $r$. This is merely a combinatorial argument: You should find that this equals, $$\dfrac{\binom{n}{r}\cdot r!}{r}$$

Centralizer of $(1234567)$:

Nextly, we describe the form of the element that commutes with this element. It should be precisely, $$(1234567)^i \sigma~;~~0\leq i < 6 ~~~\text{$\sigma$ is disjoint from $(1234567)$}$$ The facts you'll need are:

• The order of an $r$-cycle is $r$
• Disjoint cycles commute
• Cyclic groups are also abelian.

Hope this helps.

References: Herstein's Exercises and Dummit and Foote's description.

$\dagger$ This requires a proof! The reader is advised to prove this and not take it on faith because, having written this detailed answer, I'd have added this as well, if I knew the argument :)

For a finite group, there is a perfectly systematic way: to find the conjugacy class of $x$, just compute every element $g^{-1}xg$, and to find the centralizer, just compare $gx$ to $xg$ for all $g$ in the group.

For $S_n$, two elements are conjugate if and only if they have the same cycle structure. So $(123)(45)$ is conjugate to $(396)(47)$ in $S_9$, but not to $(12)(45)$ or $(12345)$ or....

There are lots of ways to use other facts you may know about a group or about an element in a group to simplify the calculation of a conjugacy class or a centralizer, but I'm afraid there is no systematic way to list all these facts. They just come with doing examples or reading worked examples in textbooks.