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pivoxa15
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Homework Statement
G=(Z+Z+Z)/N where Z denote the integers and + is direct sum and
N = <(7,8,9), (4,5,6), (1,2,3)> or the smallest submodule of Z+Z+Z containing these 3 vectors.
How would you describe G?
The Attempt at a Solution
N = {a(7,8,9)+b(4,5,6)+c(1,2,3)|a,b,c in Z} = {(7a+4b+c, 8a+5c+2c, 9a+6b+3c) | a,b,c in Z}
For any element in Z+Z+Z such as (1,1,0)
(1,1,0) + N = {(7a+4b+c+1, 8a+5b+2c+1, 9a+6b+3c)} But each component of the vector in the set must be of the form {(7a+4b+c, 8a+5c+2c, 9a+6b+3c) | a,b,c in Z}. So (1,1,0) + N = (0,1,0)+ {(7a+4b+c, 8a+5b+2c, 9a+6b+3c)| all a,b,c in Z}
Is the above correct? If it is then feeding some random vectors, i.e (,,)+N seems to always produce {(0,a,b)|a in Z2 and b in Z3}=G where Z2 is Z mod 2. Z3 is Z mod 3.
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