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Unformatted text preview: Math 4107, Midterm 1, Fall 2005 September 22, 2009 1. (10 points) a. Define what it means for a set G to be a group. b. Define what it means for a mapping from a group G to a group G to be a homomorphism. Also, define what it means for to be an isomorphism. c. Define what it means for a subgroup H of a group G to be a normal subgroup. 2. (20 points) Suppose that G is an abelian group of odd order. a. Prove that the product of all the elements of G equals the identity. b. Show that this is not true when G has even order (it is sometimes true, but not always, when G has even order) by producing a group of even order whose product of elements does not equal the identity. 3. (30 points) Suppose that G is a group of order 15. In this problem we will prove that G has an element of order 5 (in a round-about way): Break G down into orbits under conjugation, where a and b lie in the same orbit if and only if b = g 1 ag for some g G ....
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