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

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

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4.2 Ratings (578 Votes)

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

This implies that the order |?(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.
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