Commensurator
Encyclopedia
In group theory
, a branch of abstract algebra
, the commensurator of a subgroup
H of a group
G is a specific subgroup of G.
H and leave the result commensurable
with H. In other words
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a branch of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the commensurator of a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
H of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
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), is the set of all elements g of G that conjugateInner automorphism
In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
H and leave the result commensurable
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.-History of the concept:...
with H. In other words
Properties
- commG(H) is a subgroup of G.
- commG(H) = G for any compact open subgroup H.