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 $G$.

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Reflexivity^{}: we have that ${S}_{1}\sim {S}_{1}$, since $[{S}_{1}:{S}_{1}]=1$.

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Symmetry^{}: is clear from the definition.

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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:

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All nonzero subgroups of $\mathbb{Z}$ are commensurable with each other.

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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 

Canonical name  CommensurableSubgroups 
Date of creation  20130322 18:34:14 
Last modified on  20130322 18:34:14 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  4 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 20C08 
Related topic  CommensurableNumbers 
Defines  commensurable 