In mathematical finite group theory, the Dade isometry is an isometry

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

 from class functions on a subgroup H with support on a subset K of H to class function

Class function

In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function f on a group G, such that f is constant on the conjugacy classes of G. In other words, f is invariant under the conjugation map on G...

s 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).

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 K₁ is a tamely embedded subset, then the subset K consisting of K₁ without the identity element 1 satisfies the conditions above, and in this case the isometry used by Feit and Thompson is the Dade isometry.