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exam2 - MATH 4107 EXAM#2 PLEASE READ THESE DIRECTIONS...

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MATH 4107 EXAM #2 March 19, 2010 PLEASE READ THESE DIRECTIONS: Answer PROBLEM 1 (20 points) and choose TWO other problems to answer (15 points each). You may also answer (for up to 5 points extra credit) ONE additional problem. In this case, please specify which problem is the extra credit problem. All statements require proof or justification. There are 50 points total, plus up to 5 points of extra credit. 1. Let A , B be subgroups of a finite group G. (a) Let AB = { ab : a A,b B } . Prove that if A B = { e } then | AB | = | A | | B | . (b) Show that if A is a normal subgroup of G then AB is closed under compositions G. Remark: In fact, AB is a subgroup in this case, but you only have to prove that it is closed under compositions. (c) Show that if A and B are both normal then AB is normal in G. Note: You just have to prove that it is normal, the subgroup part is covered by part (b). 2. Suppose that G is a group of order 42 . (a) Show that G has a normal subgroup A of order 7 .
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