Having established the Sylow theorems, the next natural question is how they help us detect the internal shape of a finite group. One of their most powerful uses is to prove that a group is not simple. The guiding idea is direct: if Sylow congruence and divisibility force a Sylow subgroup to be unique, then that Sylow subgroup is normal, giving a nontrivial proper normal subgroup. In this lesson, we focus in the simple groups and non-simplicity tests, and the goal is to turn Sylow counting into a practical test for non-simplicity. Students often make the mistake of stopping after finding a subgroup; the subgroup must be normal before it proves non-simplicity.

Let $G$ be a group such that $G \neq \{e\}$. Then $G$ is called a \textbf{simple group} if the only normal subgroups of $G$ are $\{e\}$ and $G$.