Let $G$ be a group with identity $e_G$ and let $H$ be a group with identity $e_H$. Prove that $G$ is isomorphic to $G \oplus {e_H}$ and that H is isomorphic to ${e_G} \oplus H$.
I've defined $$g: G \to G \oplus {e_H} \; \; \text{by} \; \; g(x) = (x, e_H)$$ and $$h: H \to {e_G} \oplus H \; \; \text{by} \; \; h(x) = (e_G,x)$$
I've just learned about the external direct product and my professor wasn't available to explain it further. How do I show that my maps are 1-1 and operation preserving if I don't know the operation for $G$ and $H$? (I assume I'll be able to prove onto once I understand 1-1.)
Thanks!
As already pointed out, you should write $G\times H$ rather than using $\oplus$ as that denotes something quite different. You do not need to know the specific operations because direct product is done componentwise. That is $$(a,x)\cdot (b,x)=(a\cdot b,x\cdot y)$$ To show it is injective, or a monomorphism (which is better than 1-1 if you ask me) we simply check if the homomorphisms kernel is the trivial one. for $g$ we then have $g(x)=(e_G,e_H)$ which gives us that $x=e_G$ and as such is trivial and it is a monomorphism. Can you take it from here?