- Thread starter
- #1

- Mar 10, 2012

- 835

I observed that $(A\times B)/(A\times\{e\})\cong B$ and $(A\times C)/(A\times\{e\})\cong C$.

So I need to show that $(A\times B)/(A\times\{e\})\cong (A\times C)/(A\times\{e\})$.

Let $\psi:A\times B\rightarrow A\times C$ be an isomorphism.

Define $\phi:A\times B \rightarrow (A\times C)/(A\times\{e\})$ as $\phi(a,b)=(\psi(a,b))(A\times\{e\})$.

If I could show that $\ker \phi=A\times\{e\}$ then I'd be done.

For that I need $\psi(a,e)\in A\times\{e\}$ for all $a\in A$, which I am unable to show and this might not even be true.

Please help.