Independence and strong independence complexes of finite groups

Let be a finite group. In [10], two different concepts of independence (namely, independence and strong independence ) are introduced for the subsets of , yielding to the definition of two simplicial complexes whose vertices are the elements of . The strong independence complex turns out to be a subcomplex of the independence complex . 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.