A Classification Of Non-metabelian A₄-groups

Let p be a prime and G a finite p-group(i.e.a group of Prime Power Order).Minimal non-abelian p-groups play a key role in the study of the structure of finite p-groups.Berkovich and Janko introduced a more general concept than that of minimal non-abelian p-groups:At-group.G is called an At-group if all subgroups of index pt of G are abelian and at least one subgroup of index pt-1 of G is not abelian.Obviously,an A1-group is exactly a minimal non-abelian p-group.Also it is easy to see that for any p-group of order pn,there exists a positive integer t such that it is an At-group,where 1 ?t ?n-2.Hence the study of finite p-groups can be regarded as that of At-groups.For t<3,At-groups have been classified.The paper aims to study A4-groups.Since A3-groups have 222 non-isomorphism types,it is expected that the classification of A4-groups is extremely difficult and complex.By a observation to A3-groups we notice that they are metabelian.Thus,it seems to be reasonable to study the non-metabelian A4-groups.This thesis consists of three chapters.Chapter I is an introduction.Chapter II gives a list of preliminaries.The main contents is contained in Chapter III.We prove that if G is a non-metabelian A4-group,then p6?|G|?p9.If p=2 or 3,then non-metabelian A4 groups can be classified by using software Magma to check the groups in the SmallGroups database.So we may suppose r? 5.In this case,we can prove thst p6 ?|G| ?p8 and G'?A2.Since p-groups of order p6 and p7 were classified,we need only to consider the situation of |G|=p8.This thesis gives the classification of non-metabelian A4-groups of order p8.