# Order of Hom(Z,Z/2Z)

## Main Question or Discussion Point

I have seen that there exists two group homomorphisms from Z to Z/2Z. However, I cannot seem to understand this. I mean I know that there exists a trivial grp hom. which sends all of Z to 0 in Z/2Z. But I cannot think of anymore. Any help?

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Office_Shredder
Staff Emeritus
Gold Member
Well, Z is generated by 1. There are two places 1 can map to, 0 or 1 (mod 2). If it doesn't map to 0, it must map to 1. Do you see what function that really is?

You mean that I map all even to 0 and all odd to 1?

Landau
Yes :)
If f(1)=1 (mod 2), then f(n)=n (mod 2), which is 0 if n is even and 1 if n is odd.