Homomorphisms

How many (group) homomorphisms are there from Z20 onto (surjective to) Z8. How many are there to Z8?

Expert Solution

If φ : Z20 → Z8 is onto then there is a ∈ Z20, such that φ(a) = 1 ∈ Z8.

This implies that the order |φ(a)| is 8 and divides order of a. But the order of a divides 20.

This implies 8 divides 20, which is a contradiction. There is no homomorpphism from Z20 onto Z8.

No such homomorphisms exist from Z20 to Z8.

Nipun Maity answered 2 years ago

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