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Unformatted text preview: Final Exam, Math 4107 December 12, 2005 1. (Easy) Define the following terms: a. Group b. Ring c. Field d. Principal Ideal Domain e. Euclidean Domain 2. (Medium) In this problem you will construct a non-commutative group of order 21: We will take the set G defining our group to be Z 3 × Z 7 , and we will define a funny addition. This addition works by ( a, b ) + ( c, d ) = ( a + c (mod 3) , b · 2 c + d (mod 7)) . a. Prove that this is indeed a group. (Hint: Be careful in showing that the group operation is well-defined; also, associativity is slightly tricky. ) b. Prove that this group is non-commutative. c. (Optional +1 point extra credit) Why doesn’t this example work to show that there is a group of order 15 which is non-abelian? (We know all groups of order 15 are abelian). I could explain in about half a class where this example comes from. It is an example of a semi-direct product of Z 3 and Z 7 . Alas, the semester is over and our time has come to an end......
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