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dim&dimmer
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Homework Statement
Show that Z/mZ X Z/nZ isomorphic to Z/mnZ iff m and n are relatively prime.
(Using first isomorphism theorem)
Homework Equations
The Attempt at a Solution
Okay, first I want to construct a hom f : Z/mZ X Z/nZ ---> Z/mnZ
I have
f(1,0).m = 0(mod mn) = f(m,0) = f(e) = e and
f(0,1).n = 0(mod mn) ...
Now f(1,0) = kn for some k because f(1,0).m has to be divisible by mn
and f(0,1) = lm likewise
My hom is f(a,b) = a.f(1,0) + b.f(0,1)
Does this make sense?
If so, then I have to show that ker(f) is trivial to prove f is an iso.
ker(f)(a,b) = (0,0)
So f(a,b) = akn + blm
and I have to show akn +blm = 0.
I find this all very confusing so any help would be greatly appreciated.
Dim