# A question on group theory

#### arz2000

Hi all,

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')

Hint. An isomorphism G_n ~ G_m would lift to a map from G to G, show that this map whould have to be an isomorphism.
(re. exercise 10.2 of Representation Theory by Fulton and Harris)

Many Thanks

#### fresh_42

Mentor
2018 Award
To use the hint, write down the corresponding two short exact sequences (the groups are semidirect products!) and then formulate what "lift" means and why you get two different structures on $G$.

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