Samiksha Jaiswal (Editor)

Commensurator

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

Contents

Definition

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

c o m m G ( H ) = { g ∈ G : g H g − 1 ∩ H  has finite index in both  H  and  g H g − 1 } .

Properties

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

    • References

    Commensurator Wikipedia
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