Let T be a complete first order theory in a countable relational language L. We assume relation symbols have been added to make each formula equivalent to a predicate. Adjoin a new unary function symbol σ to obtain the language Lσ; TAut is obtained by adding axioms asserting that σ is an L-automorphism. The modern study of the model companion of theories with an automorphism has two aspects. On...

A recent result of W. Kantor followed by a work of W. Feit has rekindled interest in the longstanding conjecture of finite cyclic planes. In this paper we prove that the order of the multiplier group equals the odd part of the order of the automorphism group of a Singer group if and only if the order of the plane is 2, 3, or 8. This yields another proof for Feit's result mentioned above.

We introduce a new technique for constructing pairwise balanced designs and group divisible designs from finite groups. These constructed designs do not give designs with new parameters but our construction gives rise to designs having a transitive automorphism group that also preserves the resolution classes.

We consider the problem of uniqueness of certain simultaneity structures in flat spacetime. Absolute simultaneity is specified to be a nontrivial equivalence relation which is invariant under the automorphism group Aut of spacetime. Aut is taken to be the identity-component of either the inhomogeneous Galilei group or the inhomogeneous Lorentz group. Uniqueness of standard simultaneity in the f...

We construct two flag-transitive designs with the parameters of the title, and prove that these are the only two examples. One is pointprimitive and related to unitary geometry, while the other is pointimprimitive and constructed from a one-dimensional affine space. This classification may be contrasted with a construction by Mathon and Spence in 1996 of more than 3700 designs with these parame...

We compare Aut(Q), the classical automorphism group of a countable dense linear order, with Autc(Q), the group of all computable automorphisms of such an order. They have a number of similarities, including the facts that every element of each group is a commutator and each group has exactly three nontrivial normal subgroups. However, the standard proofs of these facts in Aut(Q) do not work for...

Dyer and Formanek (1976) proved that if N is a free nilpotent group of class two and of rank 6= 1, 3, then the automorphism group Aut(N) of N is complete. The main result of this paper states that the automorphism group of an infinitely generated free nilpotent group of class two is also complete.