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Dade isometry
In mathematical finite group theory, the Dade isometry is an isometry from class functions 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, 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'').
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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