On Generalizations Of Engel Elements

Group theory is an important branch of algebra.Many scholars have been in-vestigating on it and related topics.Thereinto,the research of Engel groups is one of important topics.In recent years,many domestic and foreign scholars have obtained some important theorem of Engel group including n-Engel groups and right(left)n-Engel groups.In this paper,we continue investigating Engel groups and the property of Engel elements,and propose the concepts of p-Engel elements,w-Engel elements and w-Engel groups,on which we carry out some research.Firstly,based on the p'-engel element and related four sets defined in[6],the p-Engel element and its four corresponding set are introduced,and their basic properties are discussed.Using p-Engel element,we obtain a sufficient or necessary condition for the finite group to be g-closed and the p-complement subgroup to be nilpotent,by the way a new subgroup and a new saturated group system are obtained.We also discuss relation between two subgroups and sets and the condition under which the critical condition holds.Secondly,it gives a generalization of the Engel group that is w-Engel group,and mainly study the equivalence relations between the right(left)n-w-Engel group,the right(left)w-Engel group and Hirsch-Plotkin radical,Baer radical,hypercenter,w-center.In addition,we obtain some properties and structures of the 2-w-Engel group.