3.II.14 F3 . \mathrm{II} . 14 \mathrm{~F} \quad

Groups, Rings and Modules | Part IB, 2004

Let LL be the group Z3\mathbb{Z}^{3} consisting of 3-dimensional row vectors with integer components. Let MM be the subgroup of LL generated by the three vectors

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

(i) What is the index of MM in LL ?

(ii) Prove that MM is not a direct summand of LL.

(iii) Is the subgroup NN generated by uu and vv a direct summand of LL ?

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

Typos? Please submit corrections to this page on GitHub.