| 词条 | Dade isometry |
| 释义 |
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 {{harv|Collins|1990|loc=6.1}}. It was introduced by {{harvs|txt|last=Dade|authorlink=Everett C. Dade|year=1964}} as a generalization and simplification of an isometry used by {{harvtxt|Feit|Thompson|1963}} in their proof of the odd order theorem, and was used by {{harvtxt|Peterfalvi|2000}} in his revision of the character theory of the odd order theorem. DefinitionsSuppose 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 → fσ from class functions f of H with support on K to class functions fσ of G, which is defined as follows: fσ(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 CG(k) is the semidirect product of a normal Hall π' subgroup I(K) with CH(k). Tamely embedded subsets in the Feit–Thompson proofThe 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
2 : Finite groups|Representation theory |
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