Thread: isomorphisms between certain groups

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

An easy way is to look at the orders of the elements, keeping in mind that isomorphisms preserve orders. In particular, the order of every element in is at most two, whereas some elements in have order four. Therefore, the two groups cannot be isomorphic.

Originally Posted by FGT12

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

To add on to roninpro's explanation it's a common fact that every group of order is abelian and thus by the structure theorem isomorphic to either or . So, just noting that narrows it down to two groups up to isomorphism. You can then extend what roninpro's one-hundred percent correct explanation to show (and I'm urging you to do this!) that if and only if !