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 $G$.

"A question on group theory"

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving