### Abstract

In this paper, we verify Dade's invariant conjecture for Steinberg's triality groups ^{3}D_{4} (2^{n}) in the defining characteristic, i.e., in characteristic 2. Together with the results in [J. An, Dade's conjecture for Steinberg triality groups ^{3}D_{4} (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 ^{3}D_{4} (2^{n}) in the defining characteristic, i.e., ^{3}D_{4} (2^{n}) is good for the prime 2 in the sense of Isaacs, Malle and Navarro.

Original language | English |
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Pages (from-to) | 802-827 |

Number of pages | 26 |

Journal | Journal of Algebra |

Volume | 316 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2007 Oct 15 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory