On the inductive McKay condition in the defining characteristic

Abstract

This note is concerned with the McKay conjecture in the representation theory of finite groups. Recently, Isaacs-Malle-Navarro have shown that, in order to prove this conjecture in general, it is sufficient to establish certain properties of all finite simple groups. In this note, we develop some new methods for dealing with these properties for finite simple groups of Lie type in the defining characteristic case. We apply these methods to show that the Suzuki and Ree groups, G 2 (q), F 4 (q) and E 8 (q) have the required properties.

On equivariant bijections relative to the defining characteristic

“…It was shown in [15] that the simple groups PSL 2 (q), 2 B 2 (2 2n+1 ), 2 G 2 (3 2n+1 ) and in [12,13] that the simple groups 2 F 4 (2 2n+1 ) and 3 D 4 (2 n ) are "good" for the defining characteristic. A uniform treatment of simple groups of Lie type with trivial Schur multiplier and cyclic outer automorphism group in the case of defining characteristic was obtained in [4]. In particular, the results in [4] include that the simple groups G 2 (q), F 4 (q) and E 8 (q) are "good" for the defining characteristic.…”

“…A uniform treatment of simple groups of Lie type with trivial Schur multiplier and cyclic outer automorphism group in the case of defining characteristic was obtained in [4]. In particular, the results in [4] include that the simple groups G 2 (q), F 4 (q) and E 8 (q) are "good" for the defining characteristic.…”

Odd character degrees for Sp(2n,2)

“…They are of degree if and only if T can be taken as T e = C G (S e ) (see [9] 6.6). Such a character is fixed by F if and only if F (T e , θ) and (T e , θ) are G F -conjugate (see [1] Section 2.1.2). This is equivalent to xF (θ) being N G (S e )-conjugate to θ ([9] 5.11).…”

“…The group G = Sp 4 (2 m ) (m 2) is simple with trivial Schur multiplier and cyclic outer automorphism group generated by F 0 (see [5]). Then the conditions of [7] Section 10 amount to find for each prime dividing |G| a proper subgroup N < G containing N G (P ) for P a Sylow -subgroup of G and such that σ (N) = N and |Irr (G) σ | = |Irr (N) σ | for any σ ∈ N Aut(G) (P ) (see [1] Section 3). The case of = 2 is also done in [1], so we assume that is odd dividing…”

Sylow d-tori of classical groups and the McKay conjecture, II

“…It is already proven that some groups are good: every simple group not of Lie type is good for all primes, dividing its order according to [9] and some groups of Lie type are good for the prime, who coincides with their defining characteristic according to [2].…”