Independence and strong independence complexes of finite groups

Let (Formula presented.) be a finite group. In [10], two different concepts of independence (namely, independence and strong independence) are introduced for the subsets of (Formula presented.), yielding to the definition of two simplicial complexes whose vertices are the elements of (Formula presented.). The strong independence complex (Formula presented.) turns out to be a subcomplex of the independence complex (Formula presented.). We discuss several invariant properties related to these complexes and ask a number of questions inspired by our results and the examples we construct. We study then the particular case of complexes on finite abelian groups, giving a characterization of the finite groups realizing them. In conclusion, answering a question of Peter Cameron, we classify all finite groups in which the two concepts of independence coincide.