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Okay, so I have to answer the following question:
How many homomorphisms are there from Z20 to Z10? How many of these are onto?
Now, I know that to find the number of homomorphisms from Z20to Z10, it suffices to see which elements of Z10 have order that is either 1, 2, 5 or 10, since these are the factors that 20 and 10 have in common. All elements of Z10 have order of either 1, 2, 5, or 10, so there are 10 homomorphisms from Z20 to Z10.
However, I'm confused about how to find out how many of these homomorphisms are onto. Does it just suffice to find phi(10), which equals 4 (and hence 4 would be the number of onto homomorphisms)?
Can I just generalize this to any homomorphism from Zm to Zn and say that assuming gcd(m, n) is not one, then the number of onto homomorphisms is always phi(n)?
A homomorphism h from Zm to Zn is onto if and only if h(1) generates Zn, or equivalently when the order of h(1) is n.
Since the order of h(1) must be a divisor of gcd(m,n), there can only be an onto homomorphism from Zm to Zn if n is a divisor of m.
If n is a divisor of m, then the number of onto homomorphisms is phi(n).
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