commensurable subgroups
0.1 Definition
Definition - Let be a group. Two subgroups are said to be commensurable, in which case we write , if has finite index both in and in , i.e. if and are both finite.
This can be interpreted informally in the following : and are commensurable if their intersection is “big” in both and .
0.2 Commensurability is an equivalence relation
- of subgroups is an equivalence relation. In particular, if and , then .
: Let , and be subgroups of a group .
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Reflexivity: we have that , since .
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Symmetry: is clear from the definition.
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0.3 Examples:
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All non-zero subgroups of are commensurable with each other.
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All conjugacy classes of the general linear group , seen as a subgroup of , are commensurable with each other.
References
- 1 A. Krieg, , Mem. Amer. Math. Soc., no. 435, vol. 87, 1990.
Title | commensurable subgroups |
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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 |