On a conjecture of Webb (Q750608)

Let \(\Delta\) be the simplicial complex of chains of non-trivial p- subgroups of a finite group G, for some prime divisor p of \(| G|\). A conjecture of Webb asserts that the orbit complex \(\Delta\) /G is contractible. In the present note, the conjecture is proved when G satisfies one of the following conditions: (a) G is p-solvable, (b) G has a Sylow p-subgroup which is either abelian, or generalized quaternion, or TI. The proofs are based on some elementary results on p-subgroups and control of fusion.