Volume 2, Issue 1

Shephard groups are common extensions of Artin and Coxeter groups. They appear, for example, in algebraic study of manifolds. An infinite family of Shephard groups which are not Artin or Coxeter groups is considered. Using techniques form small cancellation theory we show that the groups in this family are bi-automatic.

A Hurwitz group is any non-trivial finite quotient of the (2, 3, 7) triangle group, that is, any non-trivial finite group generated by elements x and y satisfying x 2 = y 3 = ( xy ) 7 = 1. Every such group G is the conformal automorphism group of some compact Riemann surface of genus g > 1, with the property that | G | = 84( g – 1), which is the maximum possible order for given genus g . This paper provides an update on what is known about Hurwitz groups and related matters, following up the author's brief survey in Bull. Amer. Math. Soc. 23 (1990).

We construct a short presentation of the ring of n × n matrices over Z with only 2 generators and 3 relations.

Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k , we prove that the diameter of said graph is asymptotically (in q ) of order . This answers a question of Benjamini. The same also holds when the generating set is taken to be a symmetric set of size 2 k .

Challenge response methods are increasingly used to enhance password security. In this paper we present a very secure method for challenge response password verification using combinatorial group theory. This method, which relies on the group randomizer system , a subset of the MAGNUS computer algebra system, handles most of the present problems with challenge response systems. Theoretical security is based on several results in asymptotic group theory.

The discrete logarithm problem is one of the backbones in public key cryptography. In this paper we study the discrete logarithm problem in the group of circulant matrices over a finite field.

In this paper we classify the coordinate ℕ-monoids of algebraic sets over the additive monoid of natural numbers.