The Congruence Subgroup Problem: An Elementary Approach Aimed at Applications

This is an elementary introduction to the congruence subgroup problem, a problem which deals with number theoretic properties of groups defined arithmetically. The novelty and, indeed, the goal of this book is to present some applications to group theory as well as to number theory which have emerged in the last fifteen years. No knowledge of algebraic groups is assumed and the choice of the examples discussed seeks to convey that even special cases give interesting applications. After the background material in group theory and number theory, solvable groups are treated first and some generalizations are presented using class field theory. Then the group SL (n) over rings of S-integers is studied. The methods involved are very different from the ones employed for solvable groups. Group theoretic properties like presentations and central extensions are extensively used. Several proofs which appeared after the original ones are discussed. The last chapter has a survey of the status of the congruence subgroup problem for general algebraic groups. Only outlines of proofs are given here and with a sufficient understanding of algebraic groups the proofs can be completed. The book is intended for beginning graduate students, many exercises are given.