# The Congruence Subgroup Problem

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*B. Sury*

A publication of Hindustan Book Agency

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.

A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels.

#### Readership

Graduate students and research mathematicians interested in algebra and algebraic geometry.