Let Z and E be two groups of all integers and even integers respectively under addition, then the mapping f from Z to E defined by; f(x)={2x for all x belongs to Z } is a homomorphism. Is it an isomorphism? Also find kernel of f?

Question

Answer

Homomorphism
- To show that the mapping f from Z to E is a homomorphism, we need to prove that f(x + y) = f(x) + f(y) for all x, y in Z.
- Let x, y be integers. Then f(x + y) = 2(x + y) = 2x + 2y = f(x) + f(y).
- Therefore, f is a homomorphism.

Isomorphism
- To determine if f is an isomorphism, we need to check if f is injective and surjective.
- Since f(x) = 2x for all x in Z, f is clearly injective.
- However, f is not surjective since E only contains even integers, while Z contains all integers.
- Therefore, f is not an isomorphism.

Kernel of f
- The kernel of f, denoted ker(f), is the set of elements in Z that map to the identity element in E.
- Since the identity element in E is 0, we need to find all x in Z such that f(x) = 0.
- This implies 2x = 0, which means x = 0.
- Therefore, ker(f) = {0}.

Similar Mathematics Doubts

Find an isomorphism from group of integer under addition to the group of even integers under addition?

Let H = Z2 x Z6 and K = Z2 x Z4, then

Let G be a non abelian group then the m apping f : G→ G given by

Which one of the following is false

Let G be a group of order 15. Then the number ofone-one homomorphism on itselfis (Answer should be integer) ___________. Correct answer is '8'. Can you explain this answer?

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams