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