The Sylow theorems are the crowning achievement of Chapter 4 and demonstrate the power of group actions. For a finite group (G) of order (p^n m) where (p) is prime and (p \nmid m):
A (left) action of a group (G) on a set (A) is a map (G \times A \to A), denoted ((g,a) \mapsto g \cdot a), such that: dummit+and+foote+solutions+chapter+4+overleaf+full
\beginproblem[Exercise 4.3.5] Show that if $G$ is a group of order $p^2$ ($p$ prime), then $G$ is abelian. \endproblem The Sylow theorems are the crowning achievement of
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