Spherical p-group complexes arising from finite groups of Lie type

We show that the p-group complex of a finite group G is homotopy equivalent to a wedge of spheres of dimension at most n if G contains a self-centralising normal subgroup H which is isomorphic to a group of Lie type and Lie rank n in characteristic p. If in addition every order-p element of G induces an inner or field automorphism on H, the p-group complex of G is G-homotopy equivalent to a spherical complex obtained from the Tits building of H. We also prove that the reduced Euler characteristic of the p-group complex of a finite group G is non-zero if G has trivial p-core and H is a self-centralising normal subgroup of G which is a group of Lie type (in any characteristic), except possibly when $p=2$ and H=Aₙ(4ᵃ) (n≥ 2) or E₆(4ᵃ). In particular, we conclude that the Euler characteristic of the p-group complex of an almost simple group does not vanish for p≥ 7.