# commensurable subgroups

## 0.1 Definition

Definition - Let $G$ be a group. Two subgroups $S_{1},S_{2}\subseteq G$ are said to be , 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

 $\displaystyle[S_{1}:S_{1}\cap S_{3}]$ $\displaystyle\leq$ $\displaystyle[S_{1}:S_{1}\cap S_{2}\cap S_{3}]$ $\displaystyle=$ $\displaystyle[S_{1}:S_{1}\cap S_{2}][S_{1}\cap S_{2}:S_{1}\cap S_{2}\cap S_{3}]$ $\displaystyle\leq$ $\displaystyle[S_{1}:S_{1}\cap S_{2}][S_{2}:S_{2}\cap S_{3}]$ $\displaystyle<$ $\displaystyle\infty.$

Similarly, we can prove that $[S_{3}:S_{1}\cap S_{3}]<\infty$ and therefore $S_{1}\sim S_{3}$. $\square$

## 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.

## References

• 1 A. Krieg, , Mem. Amer. Math. Soc., no. 435, vol. 87, 1990.
Title commensurable subgroups CommensurableSubgroups 2013-03-22 18:34:14 2013-03-22 18:34:14 asteroid (17536) asteroid (17536) 4 asteroid (17536) Definition msc 20C08 CommensurableNumbers commensurable