State the first isomorphism theorem for groups and use it to show that the groups/mz and Zm are isomorphic.
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- Exercises 35. Prove that any two groups of order are isomorphic.Suppose that is an epimorphism from the group G to the group G. Prove that is an isomorphism if and only if ker =e, where e denotes the identity in G.Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.
- Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .For a fixed group G, prove that the set of all automorphisms of G forms a group with respect to mapping composition.17. Find two groups and such that is a homomorphic image of but is not a homomorphic image of . (Thus the relation in Exercise does not have the symmetric property.) Exercise 15: 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.
- Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.Find all homomorphic images of the quaternion group.Suppose that G is a finite group. Prove that each element of G appears in the multiplication table for G exactly once in each row and exactly once in each column.
- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.Exercises 12. Prove that the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers. (Hint: Consider the mapping defined by for all .)Prove that if is an isomorphism from the group G to the group G, then 1 is an isomorphism from G to G.