Originally Posted by

**platinumpimp68plus1** I think I'm confused about notation perhaps.

let $\displaystyle p: \mathbb{Z} x \mathbb{Z} -> G$ be a homomorphism

define $\displaystyle p(m,n)=h^{m}k^{n}$ for h,k in G

give a necessary and sufficient condition for it to be a homomorphism... prove why.

i know to show something is a homomorphism i have to show that $\displaystyle p(ab)=p(a)p(b)$, ie. that the binary operation's action is preserved in both groups. but i don't know how to here. for a=(m,n) and b=(i,j), is ab=(mi,nj)? that would give $\displaystyle h^{mi}k^{nj}=h^{m+i}k^{n+j}$. How do I go about setting conditions to make these equal...? Seems to me they only would be if h,k were 0 or 1?