Having Trouble with Groups

Question

Suppose that $H$ and $K$ are groups, $G = H \times K$. $A = \{(h,1): h \in H\}$.

(a) Prove that $A$ is a normal subgroup of $G$ isomorphic to $H$.

(b) Prove that the factor group $\frac GA$ is isomorphic to $K$.

Answer 1

Here is a sketch of what needs to be done.

The map $(h,k)\mapsto K$ is a homomorphism whose kernel is $\{(h,k): k=1, h\in H\}=\{(h,1): h\in H\}=A$. (Verify these statements)

Since $A$ is the kernel of a homomorphism it is a normal subgroup of $G$.

The map from $A$ to $H$ defined by $(h,1)\mapsto h$ is the obvious isomorphism that makes $A\simeq H$. (Verify this)

$G/A$ is the set of cosets $Ag$, i.e. $(H,1)(h,k)=(H,k)$

The projection map, $(H,k)\mapsto k$, like the map above, is the isomorphism that makes $G/A\simeq K$ (Verify this)