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 (Collins 1990, 6.1). It was introduced by Dade (1964) as a generalization and simplification of an isometry used by Feit & Thompson (1963) in their proof of the odd order theorem, and was used by Peterfalvi (2000) 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.

References

Collins, Michael J. (1990), Representations and characters of finite groups, Cambridge Studies in Advanced Mathematics, vol. 22, Cambridge University Press, ISBN 978-0-521-23440-5, MR 1050762
Dade, Everett C. (1964), "Lifting group characters", Annals of Mathematics. Second Series, 79: 590–596, ISSN 0003-486X, JSTOR 1970409, MR 0160813
Feit, Walter (1967), Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, MR 0219636
Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order", Pacific Journal of Mathematics, 13: 775–1029, ISSN 0030-8730, MR 0166261
Peterfalvi, Thomas (2000), Character theory for the odd order theorem, London Mathematical Society Lecture Note Series, vol. 272, Cambridge University Press, ISBN 978-0-521-64660-4, MR 1747393