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Is it possible to generalize it?

by sukyung
Tags: generalize
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Feb17-11, 08:55 PM
P: 3
Denote Z_n=(0.1.2....n-1)

Then could I generalize the number of homomorphism H:Z_n -> Z_m as

gcd(n, m)=#(H:Z_n -> Z_m) ?

(Don't consider the case H:Z -> Z)

For example #(H: Z_4 -> Z_2)=2
#(H: Z_12 -> Z_5)= 1 (obviously the trivial one)
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Feb17-11, 11:19 PM
P: 234
Yes, and in fact [tex]Hom_\mathbb{Z}(Z_n,Z_m) \cong Z_{(n,m)}[/tex], that is, the homomorphisms from Z_n to Z_m form a cyclic group of order gcd(n,m).

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