Representations, commutative algebra, and Hurwitz groups

Representations, commutative algebra, and Hurwitz groups Dedicated to Charles Leedham-Green on the occasion of his 65th birthday

The developed methods are much more general and can in principle be used to construct representations of any finitely presented group. One assigns matrices with indeterminate entries to the generators of the group so that the group relations become relations between commuting variables, as was already suggested in [PlS 97] and applied in [HPS 97]. Meanwhile rather effective methods for treating...

Algebra Handout 2.5: More on Commutative Groups

1. Reminder on quotient groups Let G be a group and H a subgroup of G. We have seen that the left cosets xH of H in G give a partition of G. Motivated by the case of quotients of rings by ideals, it is natural to consider the product operation on cosets. Recall that for any subsets S, T of G, by ST we mean {st | s ∈ S, t ∈ T }. If G is commutative, the product of two left cosets is another left...

Commutative Algebra Commutative Algebra : General

Hurwitz Groups and Surfaces

Hurwitz not only gave an upper bound for the number of automorphisms of a compact Riemann surface of genus greater than 2, but also gave a characterization of which finite groups could be groups of automorphisms achieving this bound. In practice, however, the identification of such groups and of the surfaces they act on is difficult except in special cases. We survey what is known. 1. How I Got...

Commutative Algebra

Introduction 5 0.1. What is Commutative Algebra? 5 0.2. Why study Commutative Algebra? 5 0.3. Acknowledgments 7 1. Commutative rings 7 1.1. Fixing terminology 7 1.2. Adjoining elements 10 1.3. Ideals and quotient rings 11 1.4. The monoid of ideals of R 14 1.5. Pushing and pulling ideals 15 1.6. Maximal and prime ideals 16 1.7. Products of rings 17 1.8. A cheatsheet 19 2. Galois Connections 20 2...