commensurable subgroups

0.1 Definition

Definition - Let $G$ be a group. Two subgroups $S_1,S_2\subseteq G$ are said to be commensurable, in which case we write $S_1\sim S_2$, if $S_1\cap S_2$ has finite index both in $S_1$ and in $S_2$, i.e. if $[S_1:S_1\cap S_2]$ and $[S_2:S_1\cap S_2]$ are both finite.

This can be interpreted informally in the following : $S_1$ and $S_2$ are commensurable if their intersection $S_1\cap S_2$ is “big” in both $S_1$ and $S_2$.

0.2 Commensurability is an equivalence relation

- of subgroups is an equivalence relation. In particular, if $S_1\sim S_2$ and $S_2\sim S_3$, then $S_1\sim S_3$.

: Let $S_1$, $S_2$ and $S_3$ be subgroups of a group $G$.

• Reflexivity: we have that $S_1\sim S_1$, since $[S_1:S_1]=1$.

• Symmetry: is clear from the definition.

• Transitivity: if $S_1\sim S_2$ and $S_2\sim S_3$, then one has

  $[S_1:S_1\cap S_3]$$\le$$[S_1:S_1\cap S_2\cap S_3]$

  $=$$[S_1:S_1\cap S_2][S_1\cap S_2:S_1\cap S_2\cap S_3]$

  $\le$$[S_1:S_1\cap S_2][S_2:S_2\cap S_3]$

  $\mathrm{\infty }.$

  Similarly, we can prove that and therefore $S_1\sim S_3$. $\mathrm{\square }$

0.3 Examples:

• All non-zero subgroups of $\mathbb{Z}$ are commensurable with each other.

• All conjugacy classes of the general linear group $GL(n;\mathbb{Z})$, seen as a subgroup of $GL(n;\mathbb{Q})$, are commensurable with each other.