A graph X is k-arc-transitive if its automorphism group acts transitively on the set of it-arcs of X. A circulant is a Cayley graph of a cyclic group. A classification of 2-arc-transitive circulants is given.

The fixing number of a graph G is the cardinality of a smallest set of vertices of G that is not fixed by any non-trivial automorphism of G. In this paper we investigate the fixing number of finite graphs with abelian automorphism group.

An old conjecture of Marušič, Jordan and Klin asserts that any finite vertextransitive graph has a non-trivial semiregular automorphism. Marušič and Scapellato proved this for cubic graphs. For these graphs, we make a stronger conjecture, to the effect that there is a semiregular automorphism of order tending to infinity with n. We prove that there is one of order greater than 2.

We prove that for a polynomial f ∈ k[x, y, z] equivalent are: (1)f is a k[z]-coordinate of k[z][x, y], and (2) k[x, y, z]/(f) ∼= k[2] and f(x, y, a) is a coordinate in k[x, y] for some a ∈ k. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate f ∈ k[x, y, z] which is also a k(z)-coordinate, is a k[z]-coordinate. We discuss a method for const...

In this paper, we prove that the modular curve X(11) over a field of characteristic 3 admits the Mathieu group M 11 as an automorphism group. We also examine some aspects of the geometry of the curve X(11) in characteristic 3. In particular, we show that every point of the curve is a point of inflection, the curve has 110 hyperflexes and there are no inflectional triangles and 144 inflectional ...

A 2-(u, k, 1) block design is a set Q of v points and a collection of k-subsets of 9, called blocks, such that any two points lie in a unique block. Such designs have been classified in [ l&S] if their automorphism group acts doubly transitively or doubly homogeneously on its points. A classification for the case of flag transitive automorphism groups is now under way Cl, 2, 91. In this paper w...

In this article we study the space of leftand bi-invariant orderings on a torsion-free nilpotent group G. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of G. We prove a result which allows us to establish the same conclusion when G is assumed to be merely residually torsion-free nilpotent. In particular, we obtain faithful act...

Bremke and Xi determined the lowest two-sided cell for affineWeyl groups with unequal parameters and showed that it consists of at most |W0| left cells where W0 is the associated finite Weyl group. We prove that this bound is exact. Previously, this was known in the equal parameter case and when the parameters were coming from a graph automorphism. Our argument uniformly works for any choice of...

The automorphism group G2 of the octonions changes when octonion X,Y-product variants are used. I present here a general solution for how to go from G2 to its X,Y-product variant. * Happy Birthday to Larry Horwitz. If there’s a next time you must come.