Sylow Subgroups of Locally Finite Groups

Abstract

The theorems of Sylow are among the most basic in the theory of finite groups, and Hall’s theorems on the existence and conjugacy of Hall π-subgroups occupy a similarly central position in the theory of finite soluble groups. It is therefore natural to ask for what kinds of infinite groups results like them are true, and to what extent other parts of finite group theory can be extended to such groups. The answer to the first question has probably turned out to be more disappointing than was at one time expected, in that it is only under rather severe restrictions that sensible analogues of Sylow’s and Hall’s theorems can be obtained; but this paradoxically rekindles interest in the question in that one may now hope for a reasonably complete classification of the circumstances under which theorems like Sylow’s and Hall’s are true. As regards the second question, there are nevertheless some interesting classes of groups with very civilized Sylow structure, and quite a lot of progress has been made in extending such things as formation theory to these groups. I want mainly to discuss the first question from the classification point of view, but first some background may be of interest.