# Is it possible to generalize it?

by sukyung
Tags: generalize
 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)
 P: 234 Yes, and in fact $$Hom_\mathbb{Z}(Z_n,Z_m) \cong Z_{(n,m)}$$, that is, the homomorphisms from Z_n to Z_m form a cyclic group of order gcd(n,m).

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