On Quillen's conjecture for p-solvable groups

Quillen's conjecture and unitary groups

“…Remark 2.16. Theorem 2.14 improves the results of the earlier works [4] and [5] as explained next. There, a maximal elementary abelian p-subgroup E = e 1 , .…”

“…, i r−2 ] and either i = j or j r−1 = i r−1 . In the latter case, by [4,Lemma 2.4], sgn(i) = − sgn(j) and again the two terms cancel each other out. Finally, if k = l = r − 1, the tuples i and j are identical and the terms contribute to the sum in the statement of the lemma.…”

“…Note that in Definition 2.2, the number n of transpositions is not uniquely defined, but its parity is. For instance, if i = [1, 4, 2] ∈ S 4 , then n = 1, since we need to apply (2,4) to reorder i as [1,2,4], and m = 1, since [1,2,4] differs from [1,2,3] only in one place. Thus sgn(i) = 1.…”

“…The work by Aschbacher and Smith in [1] is so far the greatest advance towards proving the conjecture in full generality. In [4] and [5], a new geometric method is introduced and employed to prove most of the results in [1]. However, the conjecture remains open for the case of the unitary groups PSU n (q) and its p-extensions, where q is odd and p is odd and divides q + 1.…”

“…To prove this result, a non-trivial homology class in top dimension is built by gluing different conjugates of an (r − 1)-simplex, where r is the p-rank of PSU n (q). The main difference between [4] and [5] and the construction here is on the conjugating elements. In previous works these elements pairwise commuted, giving rise to a triangulation of the sphere S r−1 with 2 r (r − 1)-simplices.…”