# Dade isometry

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In mathematical [finite group theory](/source/Finite_group_theory), the **Dade isometry** is an [isometry](/source/Isometry) from [class function](/source/Class_function) on a subgroup *H* with [support](/source/Support_(mathematics)) on a subset *K* of *H* to class functions on a group *G* (Collins 1990, 6.1). It was introduced by txt as a generalization and simplification of an isometry used by Feit & Thompson (1963) in their proof of the [odd order theorem](/source/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](/source/Feit%E2%80%93Thompson_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

- Collins, Michael J. (1990), [*Representations and characters of finite groups*](https://books.google.com/books?isbn=0521234409), Vol. 22, Cambridge Studies in Advanced Mathematics, [Cambridge University Press](/source/Cambridge_University_Press), ISBN 978-0-521-23440-5. MR 1050762
- Dade, Everett C. (1964), "Lifting group characters", *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **79** (3): 590–596, Second Series, [doi:10.2307/1970409](https://doi.org/10.2307/1970409). [ISSN 0003-486X](https://www.worldcat.org/issn/0003-486X). [JSTOR 1970409](https://www.jstor.org/stable/1970409). MR 0160813
- Feit, Walter (1967), [*Characters of finite groups*](https://books.google.com/books?id=t-vuAAAAMAAJ), W. A. Benjamin, Inc., New York-Amsterdam, ISBN 9780805324341. MR 0219636
- Feit, Walter & Thompson, John G. (1963), ["Solvability of groups of odd order"](http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.pjm&issue=1103053941), *Pacific Journal of Mathematics*. **13**: 775–1029, [doi:10.2140/pjm.1963.13.775](https://doi.org/10.2140/pjm.1963.13.775). [ISSN 0030-8730](https://www.worldcat.org/issn/0030-8730). MR 0166261
- Peterfalvi, Thomas (2000), [*Character theory for the odd order theorem*](https://books.google.com/books?isbn=052164660X), Vol. 272, London Mathematical Society Lecture Note Series, [Cambridge University Press](/source/Cambridge_University_Press), [doi:10.1017/CBO9780511565861](https://doi.org/10.1017/CBO9780511565861). ISBN 978-0-521-64660-4. MR 1747393

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