202 3. PRODUCTS OF GROUPS Theorem 3.6.14. (Primary decomposition theorem) Let G be a ﬁnite abelian group and let p 1 ;:::;p s be the primes dividing j G j . Then G Š GŒp 1 Ł ± ²²² ± GŒp s Ł . Proof. Let n D p k 1 1 p k 2 s ²²² p k s s be the prime decomposition of n D j G j . Applying the previous proposition with ˛ i D p k i i gives the result. n The decomposition of Theorem 3.6.14 is called the primary decompo-sition of G . Corollary 3.6.15. (Sylow’s theorem for Finite Abelian Groups) If G is a ﬁnite abelian group, then for each prime p , the order of GŒpŁ is the largest power of p dividing j G j . Moreover, any subgroup of G whose order is a power of p is contained in GŒpŁ . Proof. By Corollary 3.6.12 , the order of GŒpŁ is a power of p . Since j G j D Q p j GŒpŁ j , it follows that j GŒpŁ j is the largest power of p dividing j G j . If A is a subgroup of G whose order is a power of p , then A is a p –group, so A ³ GŒpŁ , by deﬁnition of GŒpŁ . n The primary decomposition and the Chinese remainder theorem. For
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