Number of homomorphism from8 &oplus; 2, onto4 &oplus; 4, (Answer Should be integer) _______.Correct answer is '0'. Can you explain this answer?

Question

Answer

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.

Answer

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⊕ℤ₂contains 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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