Lattice $G_2$

Geometry Level 4

In two-dimensional Euclidean plane, let $\Lambda$ be the group generated by vectors $\lambda_1$ and $\lambda_2,\Lambda_r$ the group generated by vectors $\alpha_1$ and $\alpha_2$ (see the figure). $\Lambda=\{m\lambda_1+n\lambda_2\mid m,n\in\mathbb Z\}, \\\Lambda_r=\{m\alpha_1+n\alpha_2\mid m,n\in\mathbb Z\}.$

Clearly, $\Lambda_r$ is a subgroup of $\Lambda.$ Find the order of quotient group $\Lambda/\Lambda_r.$

3

4

\infty

1

2

1 solution

Noe Blassel
May 31, 2019

Since $\alpha_{1}=2\lambda_{1}-\lambda_{2}$ and $\alpha_{2}=2\lambda_{2}-3\lambda_{1}$ , $\Lambda_{r}=\langle \alpha_{1},\alpha_{2} \rangle_{\mathbb{Z}} \leq \langle \lambda_{1},\lambda_{2} \rangle_{\mathbb{Z}}=\Lambda \leq \mathbb{Z^2}$ . Conversely, $\lambda_{1}=2\alpha_{1}+\alpha_{2}$ and $\lambda_{2}=3\alpha_{1}+2\alpha_{2}$ , so $\Lambda \leq \Lambda_{r}$ . Hence $\Lambda=\Lambda_{r}$ , $\Lambda/\Lambda_{r}$ is trivial, and has order $1$ .

Lattice $G_2$

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Lattice G 2 G_2 G 2 ​

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Lattice $G_2$