Requesting a bit of help on the last part of my homework.
Need to prove that the following is a surjective homomorphism:
Let H and K be groups. Define the projection map:
: H x K H such that : (h, k) h
: H x K K such that : (h, k) k
How do you prove something is onto?
I know for a homomorphism you show that:
(ab) = (a)(b)
How would I attack this proof?
(hh',k) = hh' (def )
= h h'
= (h,k) (h',k) (def operation H x K)
Since (hh',k) = (h,k) (h',k) is a homomorphism
Since a homomorphism let be the identities.
(h,e) = h (def )
Therefore, a surjective homomorphism.
By a similar argument, is also a surjective homomorphism.
This feels very wrong to me...but rarely do proofs ever feel right.
Am I on the right track?