Question

Give example other than those in the text, of infinite groups G and H such that

(a) [G:H] is finite

Short answer

The index [G: H] is finite where the infinite group G and H are $\mathrm{\mathbb{Z}}\times {\mathrm{\mathbb{Z}}}_2$ and$\mathrm{\mathbb{Z}}\times \left\{0\right\}$ .

Step-by-step solution

Index of H  in  G

If H is a subgroup of a group G , then the number of distinct right cosets of H in G is called the index of H in G and it is denoted by [G : H] .

Example of  [G : H ] is finite

Let G be the infinite group $\mathrm{\mathbb{Z}}\times {\mathrm{\mathbb{Z}}}_2$ and H be the infinite subgroup of G as $\mathbb{Z}\times \left\{0\right\}$ then, there are two distinct right cosets of H which are $\left\{H,\left\{0\right\}\times {\mathrm{\mathbb{Z}}}_2\right\}$ .

Thus, there are finite number of cosets in H implies that [G : H] is finite.

Therefore, [G : H] is finite where the infinite group G and H are $\mathrm{\mathbb{Z}}\times {\mathrm{\mathbb{Z}}}_2$and $\mathrm{\mathbb{Z}}\times \left\{0\right\}$.