What Determines the Isomorphism of G/X in Group Theory?

[arz2000]

Hi,

Consider the simply connected group G of all 3 by 3 matrices
[1 a b
0 1 c
0 0 1 ]
where a,b,c are in C. The center of G is the subgroup
Z(G)={ [1 0 b
0 1 0
0 0 1] ; b is in C}
So Z(G) is isomorphic to C and therefore the discrete subgroups of Z(G) are just lattices X of rank 1 or 2.
Now show that G/X is determined up to isomorphism by the one-dimensional
Z(G)/X.

Can anybody help me solve it?

 

[matt grime]

What have you tried to do?

 

[arz2000]

In fact I don't know what should I do?

 

[matt grime]

You start with the definitions. As you always do. You look at what happens when you quotient out by X. You suppose that there are two different X's that have the same quotient Z(G)/X and try to figure out why the G/X are different. In short you think about it for a while and play around with things until you get a better idea of what's going on.

Get yourself a copy of Polya's how to solve it and read it. Your problems don't seem to have anything to do with the mathematics of this particular question, or any of the ones you post, but with not knowing how to attack problems.

 

[arz2000]

Thanks for your help.