In this paper we find a necessary and sufficient condition for a finite nilpotent group to have an abelian central automorphism group.

We give a classification of all the countable 1-transitive cyclic orderings, being those on which the automorphism group acts singly transitively. We also classify all the countable 1-transitive coloured cyclic orderings, where these are countable cyclic orderings in which each point is assigned a member of a set C, thought of as its ‘colour’, and by ‘1transitivity’ we now mean that the automor...

The automorphism group of a 3-generated 2-group G of intermediate growth is determined and it is shown that the outer group of automorphisms of G to be an elementary abelian 3group of infinite rank.

There are 50,024 Kirkman triple systems of order 21 admitting an automorphism of order 2. There are 13,280 Kirkman triple systems of order 21 admitting an automorphism of order 3. Together with the 192 known systems and some simple exchange operations, this leads to a collection of 63,745 nonisomorphic Kirkman triple systems of order 21. This includes all KTS(21)s having a nontrivial automorphi...

Let A be an infinitely generated free abelian group. We prove that the automorphism group Aut(A) first-order interprets the full secondorder theory of the set |A| with no structure. In particular, this implies that the automorphism groups of two infinitely generated free abelian groups A1, A2 are elementarily equivalent if and only if the sets |A1|, |A2| are second-order equivalent.

A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer’s 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation wit...

Suppose that an automorphism group G acts flag-transitively on a finite generalized hexagon or octagon S, and suppose that the action on both the point and line set is primitive. We show that G is an almost simple group of Lie type, that is, the socle of G is a simple Chevalley group.

A regular and edge transitive graph which is not vertex transitive is said to be semisymmetric Every semisymmetric graph is necessarily bipartite with the two parts having equal size and the automorphism group acting transitively on each of these two parts A semisymmet ric graph is called biprimitive if its automorphism group acts prim itively on each part In this paper a classi cation of bipri...

This note is part of a general programme to classify the automorphism groups of nite linear spaces. There have been a number of contributions to this programme including two recent surveys, 8, 3]. One of the most signiicant contributions was the classiication of ag-transitive linear spaces, 1]. Since then the eeort has been to classify the line-transitive examples. These fall naturally into two...

A nite C 3-geometry is called anomalous if it is neither a building nor the A 7-geometry. It is conjectured that no ag-transitive thick anomalous C 3-geometry exists. For a ag-transitive thick anomalous C 3-geometry, we prove that its 2-order y is odd and that its full automorphism group is non-solvable. As a corollary, there are no ag-transitive circular extensions of duals of anomalous C 3-ge...