If you don’t know what that means, well…you probably shouldn’t be reading this article anyway. But let’s dive right in!

First: what are conjugacy classes? Well, they’re basically just sets of elements within a group that are related to each other by some sort of transformation or “conjugation.” In the case of symmetric groups (which we’ll be focusing on today), these transformations involve swapping around elements in a specific way.

So let’s say you have this little guy:

[Insert image of a permutation]

This is called a “permutation” because it rearranges the order of some set (in this case, we’re using the numbers 1 through 5). Now imagine that we want to swap two elements in this permutation. We could do something like:

[Insert image of swapping two elements]

This new permutation is called a “conjugate” of our original one, because it’s related by some sort of transformation (in this case, we just swapped the positions of 2 and 4). And if you look closely at these two permutations, you’ll notice that they have something in common: namely, the fact that they both leave the numbers 1 and 5 alone. This is what makes them “conjugate” they share some sort of symmetry or pattern within their structure.

Now conjugacy classes specifically in symmetric groups. In these groups (which are made up of permutations), we can define a set of elements that are all related to each other by some sort of transformation. For example, if we have this permutation:

[Insert image of a permutation]

We could create a conjugacy class by swapping the positions of two elements (say, 2 and 4), like so:

[Insert image of swapping two elements in a symmetric group]

This new permutation is still part of our original set, because it’s related to it by some sort of transformation. And if we keep doing this swapping different pairs of elements within the same set we can create an entire collection of conjugacy classes that are all related to each other in some way.

So what’s the big deal about these conjugacy classes? Well, for one thing, they help us understand how permutations behave within a group (which is pretty important if you want to do any sort of math with them). They also have some interesting properties that we can use to our advantage like the fact that they’re all disjoint from each other (meaning there are no overlapping elements between different classes), and that their sizes add up to equal the size of the entire group.

But enough talk! Let’s see these conjugacy classes in action with some examples:

As you can see, each set of elements within a symmetric group is related to every other set by some sort of transformation (in this case, we just swapped the positions of two elements). And if we keep doing this creating new sets and comparing them to existing ones we can start to build up a picture of how these conjugacy classes fit together.

Conjugacy classes in symmetric groups: not as scary as they sound, right? Of course, there’s still plenty more to learn about this topic (like the fact that some sets may be “self-conjugate,” or how we can use these classes to calculate certain properties of a group). But for now, let’s just enjoy the beauty and simplicity of these little mathematical gems.

Until next time!