Cauchy's theorem gives prime-order subgroups, but normality requires more structure. The focus keyword is normal subgroups from Cauchy's theorem, because many standard arguments begin with an element of prime order and then use group actions, indices, uniqueness, and conjugation to prove normality. These methods prepare students for Sylow's normality criteria. In this lesson, the aim is not to prove every subgroup normal, but to recognize the situations where prime-order existence forces a normal subgroup.

Let $G$ be a group and let $N\leq G$. Then $N$ is called a \textbf{normal subgroup} of $G$ if $$\begin{equation*} gNg^{-1}=N \end{equation*}$$ for every $g\in G$. We write $N\trianglelefteq G$.