Article · Wikipedia archive · Last revised Jun 15, 2026

Trichotomy theorem

In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Michael Aschbacher for rank 3, and by Gorenstein and Lyons for rank at least 4. The three classes are groups of GF(2) type, groups of "standard type" for some odd prime, and groups of uniqueness type, where Aschbacher proved that there are no simple groups.

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In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Michael Aschbacher12 for rank 3, and by Gorenstein and Lyons3 for rank at least 4. The three classes are groups of GF(2) type (classified by Timmesfeld and others), groups of "standard type" for some odd prime (classified by the Gilman–Griess theorem and work by several others), and groups of uniqueness type, where Aschbacher proved that there are no simple groups.

References

References

  1. Aschbacher, Michael (October 1981). "Finite groups of rank 3. I". Inventiones Mathematicae. 63 (3): 357–402. doi:10.1007/BF01389061. ISSN 0020-9910.
  2. Aschbacher, Michael (February 1983). "Finite groups of rank 3. II". Inventiones Mathematicae. 71 (1): 51–163. doi:10.1007/BF01393339. ISSN 0020-9910.
  3. Gorenstein, Daniel; Lyons, Richard (1983). "The local structure of finite groups of characteristic 2 type". Memoirs of the American Mathematical Society. 42 (276): 0–0. doi:10.1090/memo/0276. ISSN 0065-9266.