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Chapter15The 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 \(m\), then \(G\) does not necessarily possess a subgroup of order \(n\). For example, \(A_4\) has order 12 but does not possess a subgroup of order 6. 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.