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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