The preceding lesson developed the basic non-simplicity test: force a unique Sylow subgroup and use its normality. Many finite groups of small composite order fall quickly to this method, but a few require a second idea. In this lesson, we focus in the non-simplicity of groups of small order, and the emphasis is on combining Sylow counting with actions on cosets or on Sylow subgroups. The lesson is useful because classification questions often begin by ruling out simplicity. A common student error is to count possible Sylow numbers correctly but forget to handle every remaining case.

Let $G$ be a finite group and let $H$ be a proper subgroup of $G$ with index $r$. The \textbf{index test for non-simplicity} says that if $|G|$ does not divide $r!$, then the action of $G$ on the left cosets of $H$ has a nontrivial kernel.