Commensurable subgroups

Subgroups $H,K$ of a group $G$ are commensurable if

[H:H\cap K]<\infty\quad\text{and}\quad[K:H\cap K]<\infty.

Here each bracket is the index of a subgroup. They are commensurable up to conjugacy in $G$ if $H$ and $gKg^{-1}$ are commensurable for some $g\in G$ .

Example

Inside $(\mathbb R,+)$ , the groups $\mathbb Z$ and $2\mathbb Z$ are commensurable. The groups $\mathbb Z$ and $\sqrt2\mathbb Z$ are not: their intersection is zero and has infinite index in each.

Different comparison questions

Conjugacy allows repositioning inside the given ambient group. Abstract commensurability asks only for isomorphic finite-index subgroups and need not respect a specified embedding. Arithmetic Kleinian classification uses the ambient-group, up-to-conjugacy version.