Congruence Subgroup Problem: An Elementary Approach Aimed at Applications

This is an elementary introduction to the congruence subgroup problem, a problem that 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, that 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 these 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, such as 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.