Originally Posted by

**FGT12** It is possible to show easily that $\displaystyle \mathbb{Z} / 2 $ is isomorphic with the group of units denoted $\displaystyle U_3$ that have a multiplicative inverse modulo 3 .

defined by the isomorphic map : $\displaystyle \varphi : \mathbb{Z}/2 \rightarrow \ U_3$

$\displaystyle \varphi (0) = 1 $, $\displaystyle \varphi (1) = 2$

Furthermore we can find another isomorphic map to show that $\displaystyle \phi : U_5 \rightarrow \mathbb{Z}/2 \times \mathbb{Z}/2$

I cannot really prove that the map $\displaystyle \phi : U_5 \rightarrow \mathbb{Z}/4 $ cannot be an isomorphism. However i can define the map $\displaystyle \phi (m) = m-1$ and by counter-example to prove that this is not isomorphic as

$\displaystyle \phi(4)=\phi(2) + \phi(2)$ if it is an isomorphism

but $\displaystyle = 1 + 1 = 2$ which does equal 3

However this is not very rigorous and I would like to know if there a deeper/more rigorous way of proving that there cannot be an isomorphism