The fundamental theorem of finitely generated Abelian groups says that any finitely generated Abelian group G with m=r+p generators can be uniquely decomposed into factors:
Zr X Zp1 X ... X Zpn, where r is the rank of G. Then let us take an example.
Let G be generated by two generators x, y such that x2 y3=1. What will be the unique decomposition? If I take x as the independent generator, then
G={xn|n \in Z} U {y xn|n \in Z} U {y2 xn|n \in Z} (because y3=x-2)
so G looks like Z X Z3 in this sense.
But if I take y as the independent one, then
G={yn|n \in Z} U {x yn|n \in Z} (because x2=y-3)
so G looks like Z2 X Z in this sense.
Maybe I am missing a very silly point, but what would be the correct decomposition? Or is the example wrong?
Zr X Zp1 X ... X Zpn, where r is the rank of G. Then let us take an example.
Let G be generated by two generators x, y such that x2 y3=1. What will be the unique decomposition? If I take x as the independent generator, then
G={xn|n \in Z} U {y xn|n \in Z} U {y2 xn|n \in Z} (because y3=x-2)
so G looks like Z X Z3 in this sense.
But if I take y as the independent one, then
G={yn|n \in Z} U {x yn|n \in Z} (because x2=y-3)
so G looks like Z2 X Z in this sense.
Maybe I am missing a very silly point, but what would be the correct decomposition? Or is the example wrong?