# Commensurator

﻿
Commensurator

In group theory, a branch of abstract algebra, the commensurator of a subgroup H of a group G is a specific subgroup of G.

## Definition

The commensurator of a subgroup H of a group G, denoted commG(H) or by some comm(H)[1], is the set of all elements g of G that conjugate H and leave the result commensurable with H. In other words

$\mathrm{comm}_G(H)=\{g\in G : gHg^{-1} \cap H \text{ has finite index in both } H \text{ and } gHg^{-1}\}.$[2]

## Properties

• commG(H) is a subgroup of G.
• commG(H) = G for any compact open subgroup H.

## Notes

1. ^ Onishchick (2000), p. 94
2. ^ Geoghegan (2008), p. 348

## References

• Geoghegan, Ross (2008), Topological Methods in Group Theory, Graduate Texts in Mathematics, Springer, ISBN 978-0-387-74611-1 .
• Lie groups and Lie algebras II, Encyclopaedia of mathematical sciences, Springer, 2000, ISBN 3-540-50585-7 .

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• commensurator — noun The number by which two commensurable numbers are divisible an integral number of times …   Wiktionary

• commensurator — …   Useful english dictionary

• Commensurability (mathematics) — This article is about the meaning of commensurable and derived words in mathematics. For other senses, see Commensurability (disambiguation). In mathematics, two non zero real numbers a and b are said to be commensurable if a/b is a rational… …   Wikipedia