Originally Posted by
Deveno it suffices to find a homomorphism from Z_{2}xZ_{4} onto Z_{4} whose kernel is N (do you understand why? this is "the hard part").
let's look at the cosets of Z_{2}xZ_{4}/N:
we have N = {(0,0),(1,2)}, the identity
we have (0,1) + N = {(0,1),(1,3)} (this is also (1,3) + N)
we have (0,2) + N = {(0,2), (1,0)} (which is (1,0) + N, as well)
we have (0,3) + N = {(0,3), (1,1)} (which is (1,1) + N).
so if we map (0,k) + N ---> k, it appears we will have an isomorphism.
so it looks like we want a mapping that sends (0,k) ---> k and (1,k) ---> k+2 (mod 4) (there are other possible homomorphisms).
can we combine these two into a single condition?
how about sending (k,m) ---> m + 2k (mod 4)?
now verify that f(k,m) = m + 2k (mod 4) is indeed an onto homomorphism, and that ker(f) = N.