In this paper, we verify Dade's invariant conjecture for Steinberg's triality groups 3D4 (2n) in the defining characteristic, i.e., in characteristic 2. Together with the results in [J. An, Dade's conjecture for Steinberg triality groups 3D4 (q) in non-defining characteristics, Math. Z. 241 (2002) 445-469] and [J. An, F. Himstedt, S. Huang, Uno's invariant conjecture for Steinberg's triality groups in defining characteristic, in preparation], this completes the proof of Dade's conjecture for Steinberg's triality groups. Furthermore, we show that the Isaacs-Malle-Navarro version of the McKay conjecture holds for 3D4 (2n) in the defining characteristic, i.e., 3D4 (2n) is good for the prime 2 in the sense of Isaacs, Malle and Navarro.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory