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

Trichotomy theorem

In mathematical finite group theory, the trichotomy theorem divides the simple groups of characteristic 2 type and rank at least 3 into three classes...

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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).
proved that there are no finite simple groups in the uniqueness case.