Let φ : G → H be an onto homomorphism.(a) Assume that G is abelian. Does this imply that H is abelian? Whatabout the converse?(b) What if we replaced abelian by cyclic in the above question.
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Let φ : G → H be an onto homomorphism.
(a) Assume that G is abelian. Does this imply that H is abelian? What
about the converse?
(b) What if we replaced abelian by cyclic in the above question.
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- True or False Label each of the following statements as either true or false. 2. Any two abelian groups of the same order are isomorphic.31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.True or False Label each of the following statements as either true or false. 3. Every abelian group is cyclic.
- Let G be a group. Prove that the relation R on G, defined by xRy if and only if there exist an aG such that y=a1xa, is an equivalence relation. Let xG. Find [ x ], the equivalence class containing x, if G is abelian. (Sec 3.3,23) Sec. 3.3, #23: 23. Let R be the equivalence relation on G defined by xRy if and only if there exists an element a in G such that y=a1xa. If x(G), find [ x ], the equivalence class containing x.Exercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .9. Suppose that and are subgroups of the abelian group such that . Prove that .
- 32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.True or False Label each of the following statements as either true or false. 11. The invertible elements of form an abelian group with respect to matrix multiplication.