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# Uniqueness case

In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem.

The uniqueness case covers groups G of characteristic 2 type with e(G) ≥ 3 that have an almost strongly p-embedded maximal 2-local subgroup for all primes p whose 2-local p-rank is sufficiently large (usually at least 3). Aschbacher (1983a, 1983b) proved that there are no finite simple groups in the uniqueness case.

References

Aschbacher, Michael (1983a), "The uniqueness case for finite groups. I", Annals of Mathematics. Second Series 117 (2): 383–454, doi:10.2307/2007081, ISSN 0003-486X, MR 690850

Aschbacher, Michael (1983b), "The uniqueness case for finite groups. II", Annals of Mathematics. Second Series 117 (3): 455–551, ISSN 0003-486X, JSTOR 2007034, MR 690850

Stroth, Gernot (1996), "The uniqueness case", in Arasu, K. T.; Dillon, J. F.; Harada, Koichiro; Sehgal, S.; Solomon., R., Groups, difference sets, and the Monster (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ. 4, Berlin: de Gruyter, pp. 117–126, ISBN 978-3-11-014791-9, MR 1400413

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