Let $G_1,G_2$ be groups with 2 subgroups respectively $H_1,H_2$ satisfying certain conditions, must $|G_1:H_1|=|G_2:H_2|$

Question

Let $G_1,G_2$ be groups with two subgroups respectively $H_1,H_2$ such that there is a bijection $f:G_1\rightarrow G_2$ and $f|H_1$ is a bijection between $H_1,H_2$. Must $|G_1:H_1|=|G_2:H_2|$ ?

Note: If we only require that $G_1,G_2$ have the same size and $H_1,H_2$ have the same size, then it does not follow that $|G_1:H_1|=|G_2:H_2|$. As a counterexample, set $H_1=G_1=G_2=\mathbb{Z}\times\mathbb{Z},H_2=\mathbb{Z}\times\{0\}$

Thank you

Answer 1

Take a bijection between

$H_1=\mathbb{Z}\times {0} $ and $2 \mathbb{Z} \times 2\mathbb{Z} =H_2$ then extend this bijection to a bijection of

$\mathbb{Z}\times \mathbb{Z}\rightarrow \mathbb{Z} \times \mathbb{Z}$

you can do that since in the complement there are infinite countable elements