Profinite groups of finite rank

Abstract

Abstract In this chapter we are concerned with profinite groups G for which there is an integer r such that every subgroup of G can be generated by r elements. Such groups are said to have finite rank. (This use of the term ‘rank’ should not be confused with its use in expressions such as ‘free group of rank r’.) The groups of finite rank play an important role in the theory of profinite groups and in applications to abstract group theory. After establishing the notation and some basic results, we study profinite soluble groups of finite rank and we give a number of characterizations of them.