7- Let G = U15, and let K = {1, 4}, then K is a subgroup of G. List the elements of G/K, and find the order of each element in G/K. 8 - Let G be a group and let N be a normal subgroup of G. Show that if each element in H is of finite order and each element of G/N is also of finite order then each element in G is of finite order. 9- Let R be Z12, and S = {0, 2, 4, 6, 8, 10}. Show that S is a subring of G. Is it integral domain? Justify your answer.

7- Let G = U15, and let K = {1, 4}, then K is a subgroup of G. List the elements of G/K, and find the order of each element in G/K. 8 - Let G be a group and let N be a normal subgroup of G. Show that if each element in H is of finite order and each element of G/N is also of finite order then each element in G is of finite order. 9- Let R be Z12, and S = {0, 2, 4, 6, 8, 10}. Show that S is a subring of G. Is it integral domain? Justify your answer.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 14E: Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic...

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