Quasinormal subgroups of finite p-groups

每个有限群G都是它的Sylow子群的乘积，每个素数p都有一个Sylow子群来划分G的阶。这些都是最大的子群G的顺序幂p.有限群是特别好理解模他们的西洛子群。例如，单群（没有适当的非平凡正规子群）都是已知的;有一个庞大的可解群理论（即通过扩张从阿贝尔群形成的群），其中已经发现了许多详细的结构。然而，p-群本身却不能这么说。精确的分类是不可能的，关于它们也没有什么深刻的定理。（尽管近年来通过亲p团体取得了非常惊人的进展，证明了所谓的Leedham-Green/纽曼理论。本文研究了任意有限p-群G的拟正规（简称qn）子群。它们形成了一个比G的正规子群大得多的类，并且这个想法是能够用它的qn子群来更多地描述G的结构。一个qn子群H在与每个子群K的乘法下具有置换的对称性质，即HK=KH。事实上，qn，而不是正规子群，恰恰是在群的子群格的对称性下不变的（作为一个集合）。证明了qn子群是丰富的，并且它们在包含群中的位置是正则的和可描述的。

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.