Algebra

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Math 150a: Modern Algebra Homework 5 This problem set is due Wednesday, October 31. Do problems 2.4.19, 2.5.3 (equivalently, the number of partitions), and 2.6.10(a), in addition to the following: GK1. Set arithmetic can be interesting even when the sets involved are not subgroups or cosets. The first two parts of this problem involve the vector space R 2 , which among other things is an additive group. a. Let T = { x , y , 1 x y 0 } ⊂ R 2 be the filled-in triangle with vertices at ( 0 , 0 ) , ( 1 , 0 ) , and ( 0 , 1 ) . Find T T in set arithmetic. b. Let C = { x 2 + y 2 = 1 } ⊂ R 2 be the unit circle. Find C + 3 C in set arithmetic. (Here 2 C does not mean C + C + C , but rather 3 times each element of C .) c. If G is a group, then multiplication of subsets of G is itself associative. (You are not required to prove that.) Does the semigroup of subsets of G have an identity? Which subsets have inverses? GK2. Find the centers and the conjugacy classes of the dihedral groups D 5 and D 6 . (You can do this problem any way you want, but the class lecture may be helpful.) GK3.
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