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Unformatted text preview: group of order p 3 ( p a prime) is either abelian or has center of size p . Corollary 5.4.4. Let G be a group of order p n , n > 1 . Then G has a normal subgroup f e g N G . Furthermore, N can be chosen so that every subgroup of N is normal in G . Proof. If G is nonabelian, then by the proposition, Z.G/ has the desired properties. If G is abelian, every subgroup is normal. If g is a nonidentity element, then g has order p s for some s 1 . If s < n , then h g i is a proper subgroup. If s D n , then g p is an element of order p n 1 , so h g p i is a proper subgroup. n Corollary 5.4.5. Suppose j G j D p n is a power of a prime number. Then G has a sequence of subgroups f e g D G G 1 G 2 G n D G such that the order of G k is p k , and G k is normal in G for all k ....
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- Fall '08