It is possible to show easily that

is isomorphic with the group of units denoted

that have a multiplicative inverse modulo 3 .

defined by the isomorphic map :

,

Furthermore we can find another isomorphic map to show that

I cannot really prove that the map

cannot be an isomorphism. However i can define the map

and by counter-example to prove that this is not isomorphic as

if it is an isomorphism

but

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