College Algebra Exam Review 241

College Algebra Exam Review 241 - group of order p 3 ( p a...

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5.4. GROUP ACTIONS AND GROUP STRUCTURE 251 We discovered quite early that any group of order 4 is either cyclic or isomorphic to Z 2 ± Z 2 . We can now generalize this result to groups of order p 2 for any prime p . Corollary 5.4.3. Any group of order p 2 , where p is a prime, is either cyclic or isomorphic to Z p ± Z p . Proof. Suppose G , of order p 2 , is not cyclic. Then any nonidentity el- ement must have order p . Using the proposition, choose a nonidentity element g 2 Z.G/ . Since o.g/ D p , it is possible to choose h 2 G n h g i . Then g and h are both of order p , and they commute. I claim that h g i \ h h i D f e g . In fact, h g i \ h h i is a subgroup of h g i , so if it is not equal to f e g , then it has cardinality p ; but then it is equal to h g i and to h h i . In particular, h 2 h g i , a contradiction. It follows from this that h g ih h i contains p 2 distinct elements of G , hence G D h g ih h i . Therefore, G is abelian. Now h g i and h h i are two normal subgroups with h g i \ h h i D f e g and h g ih h i D G . Hence G Š h g i ± h h i Š Z p ± Z p . n Look now at Exercise 5.4.1 , in which you are asked to show that a
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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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