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 property can be interpreted informally in the following way: $S_1$ and $S_2$ are commensurable if their intersection $S_1 \cap S_2$ is ``big'' in both $S_1$ and $S_2$ .

Commensurability is an equivalence relation

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

Proof: 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 \begin{eqnarray*} [S_1:S_1 \cap S_3] & \leq & [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]\\ & \leq & [S_1:S_1 \cap S_2][S_2:S_2 \cap S_3]\\ & < & \infty. \end{eqnarray*}Similarly, we can prove that $[S_3:S_1 \cap S_3] < \infty$ and therefore $S_1 \sim S_3$ . $\square$

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.

Bibliography

1

A. Krieg, Hecke algebras, Mem. Amer. Math. Soc., no. 435, vol. 87, 1990.