Lie Ring Methods And Its Applications In Finite P-groups

Lie ring methods is an effective method to solve some problems of finite p-groups.In this thesis,we survey the application of Lie ring methods in finite p-groups,and it is obtained by using Lie ring methods that a upper bound of the order of derived subgroup of a finite p-group whose exponent is p.The thesis consists of three sections:In Section ?,Lie ring methods is introduced briefly.In Section ?,we summarizes some applications of Lie ring methods in finite p-groups.The results are divided into the following six parts:Firstly,The relation between finite p-groups and finite dimensional linear groups;Secondly,The Wiegold conjecture of finite p-groups;Thirdly,The coclass conjecture related to the breadth of finite p-groups;Fourthly,The structure of finitgroups e p-groups with some special automorphisms;Fifthly,the existence of some normal subgroups of finite p-groups;Sixthly,The classification problems of the groups of order p6 and p7,respectively.In Section ?,we use Lie ring methods give a upper bound of the order of derived subgroup of a finite p-group G whose exponent is P1 that is |G'|?pd(dc-1-1)/2,where c is the nilpotent class of G and d is the number of elements in the minimum generating system of G.In particular,in case of c?p or G is metabelian,a more accurate upper bound of the order of the derived subgroup is given.