In mathematical
finite group theory
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
, the Dade isometry is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
from
class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjuga ...
on a subgroup ''H'' with
support on a subset ''K'' of ''H'' to class functions on a group ''G'' . It was introduced by as a generalization and simplification of an isometry used by in their proof of the
odd order theorem
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
, and was used by in his revision of the character theory of the odd order theorem.
Definitions
Suppose that ''H'' is a subgroup of a finite group ''G'', ''K'' is an invariant subset of ''H'' such that if two elements in ''K'' are conjugate in ''G'', then they are conjugate in ''H'', and π a set of primes containing all prime divisors of the orders of elements of ''K''. The Dade lifting is a linear map ''f'' → ''f''
σ from class functions ''f'' of ''H'' with support on ''K'' to class functions ''f''
σ of ''G'', which is defined as follows: ''f''
σ(''x'') is ''f''(''k'') if there is an element ''k'' ∈ ''K'' conjugate to the π-part of ''x'', and 0 otherwise.
The Dade lifting is an isometry if for each ''k'' ∈ ''K'', the centralizer ''C''
''G''(''k'') is the semidirect product of a normal Hall π' subgroup ''I''(''K'') with ''C''
''H''(''k'').
Tamely embedded subsets in the Feit–Thompson proof
The
Feit–Thompson proof of the odd-order theorem uses "tamely embedded subsets" and an isometry from class functions with support on a tamely embedded subset. If ''K''
1 is a tamely embedded subset, then the subset ''K'' consisting of ''K''
1 without the identity element 1 satisfies the conditions above, and in this case the isometry used by Feit and Thompson is the Dade isometry.
References
*
*
*
*
*{{Citation , last1=Peterfalvi , first1=Thomas , title=Character theory for the odd order theorem , publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, series=London Mathematical Society Lecture Note Series , isbn=978-0-521-64660-4 , mr=1747393 , year=2000 , volume=272, url=https://books.google.com/books?isbn=052164660X , doi=10.1017/CBO9780511565861
Finite groups
Representation theory