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Number of homomorphism from ℤ8 ⊕ ℤ2, onto ℤ4 ⊕ ℤ4, (Answer Should be integer) _______.
Correct answer is '0'. Can you explain this answer?
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Most Upvoted Answer
Number of homomorphism from8 ⊕ 2, onto4 ⊕ 4, (Answer Shoul...
Explanation:
Homomorphism:
Homomorphism is a structure-preserving map between two algebraic structures of the same type. In this case, we are looking for homomorphisms between two direct sum groups.
Direct Sum:
The direct sum of two groups A and B, denoted as A ⊕ B, is the set of all pairs (a, b) where a belongs to group A and b belongs to group B. Addition in the direct sum is defined component-wise.
Onto Homomorphism:
An onto homomorphism is a homomorphism that covers the entire target group. In other words, every element in the target group has a pre-image in the source group.
Explanation of the Problem:
We are given two direct sum groups: 8 ⊕ 2 and 4 ⊕ 4. The number of homomorphisms from 8 ⊕ 2 onto 4 ⊕ 4 is zero. This is because the size of the source group (8 ⊕ 2) is 10, while the size of the target group (4 ⊕ 4) is 8. Since the target group has fewer elements than the source group, it is not possible to have an onto homomorphism.
Therefore, the number of homomorphisms from 8 ⊕ 2 onto 4 ⊕ 4 is 0.
Homomorphism:
Homomorphism is a structure-preserving map between two algebraic structures of the same type. In this case, we are looking for homomorphisms between two direct sum groups.
Direct Sum:
The direct sum of two groups A and B, denoted as A ⊕ B, is the set of all pairs (a, b) where a belongs to group A and b belongs to group B. Addition in the direct sum is defined component-wise.
Onto Homomorphism:
An onto homomorphism is a homomorphism that covers the entire target group. In other words, every element in the target group has a pre-image in the source group.
Explanation of the Problem:
We are given two direct sum groups: 8 ⊕ 2 and 4 ⊕ 4. The number of homomorphisms from 8 ⊕ 2 onto 4 ⊕ 4 is zero. This is because the size of the source group (8 ⊕ 2) is 10, while the size of the target group (4 ⊕ 4) is 8. Since the target group has fewer elements than the source group, it is not possible to have an onto homomorphism.
Therefore, the number of homomorphisms from 8 ⊕ 2 onto 4 ⊕ 4 is 0.
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Community Answer
Number of homomorphism from8 ⊕ 2, onto4 ⊕ 4, (Answer Shoul...
Clearly ℤ8 ⊕ ℤ2 = ℤ4 ⊕ ℤ4
If there exist an onto homomorphism from ℤ8⊕ℤ2, to ℤ4⊕ℤ4, then ℤ8⊕ℤ2≈ℤ4⊕ℤ4
But this is not possible since ℤ8⊕ℤ2 contains an element of order 8 but ℤ4⊕ℤ4 does not have.
⇒ There is no onto homomorphism between groups ℤ8⊕ℤ2 and ℤ4⊕ℤ4.
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Number of homomorphism from8 ⊕ 2, onto4 ⊕ 4, (Answer Should be integer) _______.Correct answer is '0'. Can you explain this answer?
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