Abstract Algebra: Homomorphisms?

Question

How do I show that

phi: Z_12 ===> Z_10

defined by phi(x) = 3x is not a homomorphism?

Be as specific as you can, please. Thanks for the help!

Answer 1

It's not clear to me what the map is here, exactly--by which I mean, the expression "φ(x)=3x" isn't really well-defined.

For example, if I take x=1, then φ(x)=3. But if I take x=13, then φ(x)=39=9. Note that 1=13 (mod 12), but 3≠9 (mod 10). This is a problem!

Leaving that aside for a moment, it's clear anyway that this thing isn't a homomorphism. Recall that the key property of a homomorphism is that φ(x+y) = φ(x) + φ(y), and that an immediate consequence of this is that φ(0)=0.

Let x=y=6; then
φ(x+y)=φ(12)=φ(0)=0, but
φ(x)+φ(y) = φ(6) + φ(6) = 18+18=6 (mod 10), and 6≠0 (mod 10).

Therefore, φ isn't a homomorphism--and it really isn't even a well-defined function anyway!
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