A Criterion for Subnormality and Wielandt Complexes in finite Groups

The main aim of this paper is to prove the following criterion for subnormality in finite groups. MAIN THEOREM. Let G be a finite group, and H a subgroup of G. Suppose that for every prime p dividing the order of G there exists a Sylow p-subgroup Gp of G such that H is subnormal in 〈H, Gp〉. Then H is subnormal in G. The proof of the theorem, which requires the classification of the finite simple groups is postponed to Section 3. Before, in Section 1, we give an application of it by proving (Theorem 2.1) that a subgroup of a finite group is subnormal if and only if the Euler characteristic of a certain simplicial complex associated to it is 1.