Quasinormal subgroups of finite p-groups

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Every finite group G is the product of its Sylow subgroups, one for each prime p dividing the order of G. These are the maximal subgroups of G of order a power of p. Finite groups are particularly well understood modulo their Sylow subgroups. For example, the simple groups (with no proper non-trivial normal subgroups) are all known; and there is a vast theory of soluble groups (i.e. those formed from abelian groups via extensions), where much detailed structure has been discovered. The same cannot be said of the p-groups themselves, however. A precise classification is out of the question and there are few deep theorems about them. (Though in recent years very striking progress has been made via pro-p-groups, proving the so-called Leedham-Green/Newman conjectures.) The project described here will investigate the quasinormal (qn for short) subgroups of an arbitrary finite p-group G. They form a significantly larger class than the normal subgroups of G, and the idea is to be able to say more about the structure of G in terms of its qn subgroups. A qn subgroup H possesses the symmetrical property of permuting under multiplication with every subgroup K, i.e. HK=KH. In fact the qns, not the normal subgroups, are precisely the ones that are invariant (as a set) under the symmetries of the group's lattice of subgroups. It is conjectured that qn subgroups are plentiful and that their situation within the containing group is of a regular and describable form.