Quillen's conjecture in rational homology at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Quillen's conjecture in rational homology

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Quillen's conjecture in rational homology. Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.

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LEVEL 1 Rational homology and Quillen's conjecture

released 2026-09-24 | 5 theorems · 23 lemmas · 29 proofs · 31,304 words | PLAY LEVEL 1 » (pdf)

We prove Quillen's conjecture for all finite groups and all primes. More precisely, if a finite group G has trivial largest normal p-subgroup $O_p(G)$, then the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology. This establishes the stronger rational-homology form of the conjecture.

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