turkmath.org

I will survey some of the known results on finite group actions on finite or finite dimensional CW-complexes homotopy equivalent to a sphere. One of the key ingredients for the construction of a G-action on a homotopy sphere is the existence of an effective representation of a Sylow p-subgroup of G which respects fusion in G. Recently we showed that these representations can be found using Glauberman’s ZJ-theorem in finite group theory. The condition for this theorem is related to the group Qd(p), and because of this, we also prove a result about groups which involve Qd(p). In the talk I will present some of ideas behind our proofs. This is a joint work with Yasir Kızmaz and the project was supported by Tubitak 1001 project with grant no 116F194.