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Problem (Introduction To Internal Direct Products)

Consider the group $G\times H$. Let $A=G\times \{e_H\}$ and $B=\{e_G\}\times H$. Prove the following.

  1. $A\cap B$ contains only the identity element of $G\times H$.
  2. The set product $AB$ equals the whole group $G\times H$.
  3. $A$ and $B$ are normal subgroups of $G\times H$.
  4. $G\approx A$ and $H\approx B$, which means $G\times H=AB\cong A\times B$.

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