In my Abstract Algebra course (MATH 403), we covered a proof that particularly confused me. If I’m being honest, I definitely zoned out while my professor was lecturing on this, and thankfully we weren’t tested on it. However, I was still curious about the proof, so I decided to revisit it and try to understand it. I’m going to try to explain it here, and hopefully it will make sense to others who were confused by it as well.

We’ll define some terms first.

Definition. The symmetric group of degree $n$, denoted by $S_n$, consists of all permutations of the set $\{1, 2, \dots, n\}$.

Definition. The alternating group of degree $n$, denoted by $A_n$, consists of all even permutations of the set $\{1, 2, \dots, n\}$.

Showing that $A_n$ is a group for any $n \geq 1$ is a good exercise, but I won’t go into it here. The important thing to note is that $A_n$ is a subgroup of $S_n$.

Definition. A group $G$ is simple if the only normal subgroups of $G$ are the trivial group $\{e\}$ and $G$ itself.

The theorem is as follows:

Theorem. For $n > 4$, the group $A_n$ is simple.