15: The Sylow Theorems
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.