Trichotomy theorem

Not to be confused with Law of trichotomy.

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 Aschbacher (1981, 1983) for rank 3 and by Gorenstein & Lyons (1983) 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

• Aschbacher, Michael (1981), "Finite groups of rank 3. I", Inventiones Mathematicae, 63 (3): 357–402, Bibcode:1981InMat..63..357A, doi:10.1007/BF01389061, ISSN 0020-9910, MR 0620676
• Aschbacher, Michael (1983), "Finite groups of rank 3. II", Inventiones Mathematicae, 71 (1): 51–163, Bibcode:1983InMat..71...51A, doi:10.1007/BF01393339, ISSN 0020-9910, MR 0688262
• Gorenstein, D.; Lyons, Richard (1983), "The local structure of finite groups of characteristic 2 type", Memoirs of the American Mathematical Society, 42 (276): vii+731, doi:10.1090/memo/0276, ISBN 978-0-8218-2276-0, ISSN 0065-9266, MR 0690900