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Just took my final exam and I wanted to see if I answered this correctly:

If $A$ is a Abelian group generated by $\left\{x,y,z\right\}$ and $\left\{x,y,z\right\}$ have the following relations:

$7x +5y +2z=0; \;\;\;\; 3x +3y =0; \;\;\;\; 13x +11y +2z=0$

does it follow that $A \cong Z_{3} \times Z_{3} \times Z_{6}$ ?

I know if we set $x=(1,0,2)$, $y=(0,1,0)$ and $z=(2,1,5)$ then this is consistent with the relations and with $A \cong Z_{3} \times Z_{3} \times Z_{6}$

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    That is not correct,2012-12-08
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    With your $x,y,z$ I get $7x+5y+2z=(11,9,16)$; how is that $0$?2012-12-08
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    Gerry you are correct. My apologies. I've made a correction for $z$2012-12-09
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    @jojo , it still is wrong: just behold the first coordinate: you get $\,9\,$ , not zero!2012-12-09
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    9 mod 3 is 0 isn't it?2012-12-09

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