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apr06 - Math 4124 Wednesday April 6 April 6 Ungraded...

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Math 4124 Wednesday, April 6 April 6, Ungraded Homework Exercise 5.2.1 on page 165. In each of parts (a) to (d), give the number of nonisomorphic abelian groups of the specified order. (a) 100 (b) 576 (c) 1155 (d) 42875 (a) First write 100 as a product of prime factors; thus 100 = 2 2 5 2 . The number of partitions of 2 is 2, namely { 2 } and { 1,1 } , thus the number of isomorphism classes of abelian groups of order p 2 is 2. It follows that the number of isomorphism classes of groups of order 100 is 2 · 2 = 4. (b) 576 = 2 6 3 2 . The partitions of 6 are { 6 } , { 5,1 } , { 4,2 } , { 4,1,1 } , { 3,3 } , { 3,2,1 } , { 3,1,1,1 } , { 2,2,2 } , { 2,2,1,1 } , { 2,1,1,1,1 } , { 1,1,1,1,1,1 } ; thus there are 11 partitions of 6. Also there are 2 partitions of 2. Therefore the number of isomorphism classes of abelian groups of order 576 is 2 · 11 = 22. (c) 1155 = 3 · 5 · 7 · 11. The number of partitions of 1 is 1. Therefore the number of isomor- phism classes of abelian groups of order 1155 is 1 · 1 · 1 · 1 = 1. (d) 42875 = 5 3 7 3 . The partitions of 3 are { 3 } , { 2,1 } , { 1,1,1 } , so there are 3 partitions of 3. Therefore the number of isomorphism classes of abelian groups of order 42875 is 3 · 3 = 9.
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