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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?