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Problem (Collapsing A Factor Of An External Direct Product Yields The Other Factor, or $(G\times H)/H\approx G$)

Let $G$ and $H$ be groups, and consider the projection map $\pi_1:G\times H\to G$ defined by $\pi_1(g,h)=g$. We call this the projection map $\pi_1$ because it takes the element $(g,h)$ and returns the first component. The projection map $\pi_2$ would return the second component $h$.

  1. Prove that this projection map is a surjective homomorphism.
  2. What is the kernel of $\pi_1$, in other words what is $f^{-1}(e_G)$?
  3. Why is $(G\times H)/(\{e_G\}\times H)\approx G$?
  4. Show that $i_H:\{e_G\}\times H\to H$ defined by $i_H(e_G,h)=h$ is an isomorphism.
Because of the work above, we'll often write $(G\times H)/H\approx G$, even though $H$ is not technically a subgroup of $G\times H$, rather $\{e_g\}\times H$ is a subgroup of $H$. However, it's ugly to write $(G\times H)/(\{e_G\}\times H)\approx G$ and much prettier to just write $(G\times H)/H\approx G$, which we justify doing because $H\approx \{e_G\}\times H$.

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