Label each of the following statements as either true or false.
Every homomorphism is an isomorphism.
Whether the statement, “Every homomorphism is an isomorphism” is true or false.
Answer to Problem 1TFE
Solution:
The statement, “Every homomorphism is an isomorphism” is false.
Explanation of Solution
Formula used:
1) Homomorphism: Let
2) Isomorphism: Let
1.
(bijective) and
2.
(homomorphism).
Explanation:
Consider the statement, “Every homomorphism is an isomorphism.”
Counter example:
Let
Mapping
First to check whether
Thus,
To check whether
As
Therefore,
So,
Thus
Hence, the statement “Every homomorphism is an isomorphism” is false.
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Chapter 3 Solutions
ELEMENTS OF MODERN ALGEBRA
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- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning