Quotient groups of direct products: $\left(G\times H\right)/G^*\cong H$ and vice versa. 
If $G$ and $H$ are groups. Let $G^\star= \{(a, e_H)| a\in G\}$ and $H^\star=\{(e_H, b) |b \in H\}$. Show that 
  $(G \times H)/G^\star$ is isomorphic to $H$ and $(G \times H)/H^\star$ is isomomorphic to $G$.

Guys I am stuck! Please help me as to what function should I use and how should I proceed?
 A: Hint: Apply the first isomorphism theorem to the projections from $G \times H$ onto $G$ and $H$.
A: Given a group $G\times H$, I find it helpful to use the following notation: $$G\times 1 = \{(g,1)\in G \times H\}.$$ You might notice an obvious isomorphism $(g,1)\mapsto g$ between $G\times 1$ and $G$.  Similarly, writing $$1\times H = \{(1,h)\in G \times H\}$$
we see an isomorphism $(1,h)\mapsto h$ between $1\times H$ and $H$.
Now, you might notice that we could have instead defined a map from $G\times H$ to $G$ by $(g,h)\mapsto g$, or a similar map from $G\times H$ to $H$ by $(g,h)\mapsto h$.  These aren't bijective, but perhaps we can use them anyway.
$\hspace{22pt}{\rm \text{S}{\small \text{TEP }} 1.}$  Verify that $(g,h)\mapsto g$ and $(g,h)\mapsto h$ are homomorphisms.
$\hspace{22pt}{\rm \text{S}{\small \text{TEP }} 2.}$  What are the kernels of $(g,h)\mapsto g$ and $(g,h)\mapsto h$?  What are their images?

 $(g,h)$ is in the kernel of $(g,h)\mapsto g$ whenever $g=1$, so the set of such $(g,h)$ is$\ldots$

$\hspace{22pt}{\rm \text{S}{\small \text{TEP }} 3.}$  Can you compose the canonical homomorphisms given by the first isomorphism
$\hspace{22pt}$theorems with any of the other maps we've seen recently?
A: Hint: What are the cosets of $G^\star$ in $G\times H$?
