Question

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

(b) [G:H] is infinite

Short answer

The index [G:H] is infinite where the infinite group G and H are $\mathrm{ℝ}$ and $\mathrm{\mathbb{Z}}$.

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 infinite

Let G be the infinite group $\mathrm{ℝ}$ and H be the infinite subgroup $\mathrm{\mathbb{Z}}$ of G then, there exists a coset $\left\{\text{x+}\mathbb{\text{ℤ}}\text{:x∈}\mathbb{\text{ℝ}}\right\}$which is infinite.

Thus, the coset $\left\{\text{x+}\mathbb{\text{ℤ}}\text{:x+}\mathbb{\text{ℝ}}\right\}$is infinite implies that [ G:H ] is infinite.

Therefore, [G:H] is infinite where the infinite group G and H are $\mathrm{ℝ}$and $\mathrm{\mathbb{Z}}$.