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Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.7: Direct Sums (optional)

Problem 14E: 14. Let be an abelian group of order where and are relatively prime. If and , prove that .

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5. Let p be a prime. Prove that the group (x, y|xP = yP = (xy)P = 1> is infinite if
p> 2, but that if p = 2, it is a Klein 4-group.

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9. Suppose that and are subgroups of the abelian group such that . Prove that .
14. Let be an abelian group of order where and are relatively prime. If and , prove that .
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
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