Let $L$ be the group $\mathbb{Z}^{3}$ consisting of 3-dimensional row vectors with integer components. Let $M$ be the subgroup of $L$ generated by the three vectors

$$u=(1,2,3), v=(2,3,1), w=(3,1,2) \text {. }$$

(i) What is the index of $M$ in $L$ ?

(ii) Prove that $M$ is not a direct summand of $L$.

(iii) Is the subgroup $N$ generated by $u$ and $v$ a direct summand of $L$ ?

(iv) What is the structure of the quotient group $L / M$ ?