# Trichotomy theorem

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Not to be confused with [Law of trichotomy](/source/Law_of_trichotomy).

In [group theory](/source/Group_theory), the **trichotomy theorem** divides the [finite simple groups](/source/Finite_simple_group) of [characteristic 2 type](/source/Characteristic_2_type) and [rank](/source/Rank_(group_theory)) at least 3 into three classes. It was [proved](/source/Mathematical_proof) by Michael Aschbacher[1][2] for rank 3, and by Gorenstein and Lyons[3] for rank at least 4. The three classes are [groups of GF(2) type](/source/Groups_of_GF(2)_type) (classified by Timmesfeld and others), groups of "standard type" for some [odd](/source/Parity_(mathematics)) [prime](/source/Prime_number) (classified by the [Gilman–Griess theorem](/source/Gilman%E2%80%93Griess_theorem) and work by several others), and [groups of uniqueness type](/source/Uniqueness_case), where Aschbacher proved that there are no [simple groups](/source/Simple_group).

## References

1. Aschbacher, Michael (October 1981). ["Finite groups of rank 3. I"](http://link.springer.com/10.1007/BF01389061). *Inventiones Mathematicae*. **63** (3): 357–402. [doi:10.1007/BF01389061](https://doi.org/10.1007/BF01389061). [ISSN 0020-9910](https://www.worldcat.org/issn/0020-9910)

1. Aschbacher, Michael (February 1983). ["Finite groups of rank 3. II"](http://link.springer.com/10.1007/BF01393339). *Inventiones Mathematicae*. **71** (1): 51–163. [doi:10.1007/BF01393339](https://doi.org/10.1007/BF01393339). [ISSN 0020-9910](https://www.worldcat.org/issn/0020-9910)

1. Gorenstein, Daniel & Lyons, Richard (1983). ["The local structure of finite groups of characteristic 2 type"](http://www.ams.org/memo/0276). *Memoirs of the American Mathematical Society*. **42** (276): 0–0. [doi:10.1090/memo/0276](https://doi.org/10.1090/memo/0276). [ISSN 0065-9266](https://www.worldcat.org/issn/0065-9266)

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