A criterion for subnormality and Wielandt complexes in finite groups (Q1337443)

This paper illustrates the power of the classification of finite simple groups to problems in group theory which appear to have no connection with finite simple groups. The main theorem is the following. Let \(G\) be a finite group and \(H\) a subgroup of \(G\). Suppose that for each prime \(p\) dividing the order of \(G\) there is a Sylow \(p\)-subgroup of \(G\), \(G_ p\), such that \(H\) is subnormal in \(\langle G_ p, H\rangle\). Then \(H\) is a subnormal subgroup of \(G\). The proof depends on reducing the problem to one concerning finite simple groups. The author then uses this theorem to consider a problem of subnormality in infinite groups related to the Wielandt complex.