Concept explainers
To find: An isomorphism from to group of integers under addition to the group of even integers under addition.
Explanation of Solution
Given information:
Concept used:
Isomorphism: - A homomorphism
Isomorphic groups: - When an isomorphism of G onto
One-one function: - A function which maps distinct elements from the domain to the distinct elements of the co-domain is said to be one-one function.
Onto function: - A function
Calculation:
Construct a function
Let us consider two elements as
Let
Then from the defined mapping
So,
Hence it is clear that the function is one-one.
Now consider
So, f is onto.
Now
Hence f is a group isomorphism and hence
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Chapter 6 Solutions
EBK CONTEMPORARY ABSTRACT ALGEBRA
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- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,