# Prove these groups are not isomorphic

## Homework Statement

Consider the groups G_n= {(a,b) in (C*) * C ; (a,b).(a',b')=( aa', b+(a^n)b' )}
where n is in N. Show that if n and m are distinct, then the two groups G_n and G_m are not isomorphic.

Ps. In fact G_n = G/ nZ , where G is the group of pairs (t,s) in C * C with group law (t,s). (t',s')=(t+t',s+(e^t)s')

## The Attempt at a Solution

Suppose in countrary that for distinct n and m there is an isomorphism from G_n onto G_m ;
f:G_n ---> G_m by g (nZ)---> g'(mZ) for all g in G.
This isomorphism authomatically leds to an isomorphism from G onto G given by h:G--->G by g--->g' for all g in G.
Now, how can I get a contradiction?