Abstract Algebra: Theory and Applications

We already know that the converse of Lagrange’s Theorem is false. If $G$ is a group of order $m$ and $n$ divides $m\text{,}$ then $G$ does not necessarily possess a subgroup of order $n\text{.}$ For example, $A_4$ has order $12$ but does not possess a subgroup of order $6\text{.}$ However, the Sylow Theorems do provide a partial converse for Lagrange’s Theorem—in certain cases they guarantee us subgroups of specific orders. These theorems yield a powerful set of tools for the classification of all finite nonabelian groups.