Given f: GXH--G f(x, y) = x and h: GXH--H h(x, y) = y show f and h are homomorphism of the group's GXH, G, and H.

Question

So I want to say f((x, y)(x', y')) = xx'.

Is it safe to assume (a, b)(c, d) = (ac, bd)? The book I using often doesn't state what the operations are. Can I assume the operation is the same for all groups or the Gs and Hs?

So f((a, b)(c, d)) = ac, f(a, b)f(c, d) = ac.

And thus f((a, b)(c, d)) = f(a,b)f(c,d).

Answer 1

It is true that $(a,b) (c,d) = (ac, bd)$. To justify this, refer back to the definition of the direct product of groups.