SUBGROUPS WHOSE GENERATORS SATISFY SOME COMMUTATOR CONDITIONS

Abstract

The main purpose of the thesis was to establish the nilpotence of subgroups whose generators satisfied some Engel conditions. Consideration was given primarily to subgroups of finite soluble groups whose generators were p-elements. Chapter III and Chapter IV were devoted mainly to the study of subgroups generated by elements which Engel elements with respect to one another. In chapter III, the subgroups considered were assumed to have Sylow p-subgroups of nilpotent class at most 2. It was proved, for example, that in soluble groups whose Sylow p-subgroups are of nilpotent class at most 2, subgroups generated by two p-elements which are Engel elements with respect to each other are nilpotent. A counterexample was given to show that this result does not hold in subgroups generated by more than two p-elements. In Chapter IV we relaxed the assumptions on the Sylow p-subgroups and imposed instead some conditions on the generators. Two types of conditions were considered. For example, it was shown that a soluble group G generated by two p-elements a and b, where a is a left Engel element with respect to be is nilpotent if the centralisers of a in all homomorphic images of G are nilpotent. In the last chapter, some commutator conditions involving the orders of the commutators were considered. It was shown that subgroups generated by some set of p-elements satisfying such commutator conditions are nilpotent. In this case, the generators were Engel elements with respect to one another. The equivalence of some commutator conditions were also established.