commensurable subgroups

0.1 Definition

Definition - Let G be a group. Two subgroupsMathworldPlanetmathPlanetmath S1,S2G are said to be commensurableMathworldPlanetmathPlanetmath, in which case we write S1S2, if S1S2 has finite index both in S1 and in S2, i.e. if [S1:S1S2] and [S2:S1S2] are both finite.

This can be interpreted informally in the following : S1 and S2 are commensurable if their intersectionMathworldPlanetmath S1S2 is “big” in both S1 and S2.

0.2 Commensurability is an equivalence relation

- of subgroups is an equivalence relationMathworldPlanetmath. In particular, if S1S2 and S2S3, then S1S3.

: Let S1, S2 and S3 be subgroups of a group G.

  • ReflexivityMathworldPlanetmath: we have that S1S1, since [S1:S1]=1.

  • SymmetryPlanetmathPlanetmath: is clear from the definition.

  • Transitivity: if S1S2 and S2S3, then one has

    [S1:S1S3] [S1:S1S2S3]
    = [S1:S1S2][S1S2:S1S2S3]
    < .

    Similarly, we can prove that [S3:S1S3]< and therefore S1S3.

0.3 Examples:

  • All non-zero subgroups of are commensurable with each other.

  • All conjugacy classesMathworldPlanetmathPlanetmath of the general linear groupMathworldPlanetmath GL(n;), seen as a subgroup of GL(n;), are commensurable with each other.


  • 1 A. Krieg, , Mem. Amer. Math. Soc., no. 435, vol. 87, 1990.
Title commensurable subgroups
Canonical name CommensurableSubgroups
Date of creation 2013-03-22 18:34:14
Last modified on 2013-03-22 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