Glauberman-Isaacs Correspondence and Π-Brauer Characters
✍ Scribed by J.S. Graves
- Publisher
- Elsevier Science
- Year
- 1994
- Tongue
- English
- Weight
- 400 KB
- Volume
- 169
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
Suppose that a group (A) acts on a group (G) of coprime order; then the Glauberman-Isaacs correspondence defines a bijection between the (A)-invariant irreducible characters of (G) and the irreducible characters of the fixed-point subgroup (C=\mathrm{C}{6}(A)). For a set of primes (\pi), and a (\pi)-separable group (G), the correspondent of an (\boldsymbol{A})-invariant (\pi)-Brauer character (\phi) was defined as follows: find (x \in B{\pi}(G) \subseteq \operatorname{Irr}(G)) which is the canonical lift for (\phi), then take the correspondent of (x), and finally restrict the correspondent of (x) to the (\pi)-elements of (C). One of the main results of this paper is to show that the correspondent of an (A)-invariant (\pi)-Brauer character is obtained if one chooses any of its (A)-invariant lifts in (\operatorname{Ir}(G)) and applies the above algorithm. Thus, in the case where (x) lifts a (\pi)-Brauer character, we can say that application of the correspondent map commutes with application of the map which restricts characters to (\pi)-elements of the group. We show by example that these maps do not commute when (x) is the lift of a sum of (A)-invariant (\pi)-Brauer characters, and prove a theorem characterizing this behavior when (A) is solvable. 1 19y4 Academic Press, inc