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- Apr 14, 2013

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Let $(G, \#), \ (H, \square )$ be groups. Show:

- For $(g,h), (g',h')\in G\times H$ we define the operation $\star$ on $G\star H$ as follows:

\begin{equation*}\star: (G\times H)\times (G\times H,\star), \ \left ((g,h), (g',h')\right )\mapsto (g\# g', h\square h')\end{equation*} Then $(G\times H, \star)$ is a group. - The map $G\rightarrow G\times H, \ g\mapsto (g, e_H)$ is a monomorphism, where $e_H$ is the neutral element in $H$.
- The map $G\times H\rightarrow G, \ (g,h)\mapsto g$ is an endomorphism.
- Let $G=\mathbb{Z}/4\mathbb{Z}$ and $H=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. Does it hold that $G\sim H$ ?

For 1:

We have to show that the four axioms (closure, associativity, identity, inverse) are satisfied.

For 2:

Let $\phi$ be the name of that map. Then we have that $\phi (g\#g')=(g\#g', e_H)$. How could we continue?

For 3:

Let $\psi$ be the name of that map. Then we have that $\psi ((g,h)\star (g',h'))=(g\#g', h\square h')$. How could we continue?

For 4:

We have to define $f:G\rightarrow H$ and check if this map is bijective, right?