What is the concept of a group homomorphism and how does it relate to the concept of isomorphism in abstract algebra?
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What is the concept of a group homomorphism and how does it relate to the concept of isomorphism in abstract algebra?
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- 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.Find two groups of order 6 that are not isomorphic.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 .)
- 27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.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.Consider the groups given in Exercise 12. Find an isomorphism from the multiplicative group + of positive real numbers to the additive group of real numbers. Prove that the additive group of real numbers is isomorphic to the multiplicative group + of positive real numbers. (Hint: Consider the mapping :+ defined by (x)=10x for all x.)
- Prove that any group with prime order is cyclic.Exercises 30. For an arbitrary positive integer, prove that any two cyclic 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.