Dave Witte

Transitive Permutation Groups of Prime-Squared Degree

AbstractWe explicitly determine all of the transitive groups of degree p2, p a prime, whose Sylow p-subgroup is not isomorphic to the wreath product

Cayley digraphs of prime-power order are Hamiltonian

\mathbb{Z}_p \wr \mathbb{Z}_p

The Hamilton spaces of Cayley graphs on abelian groups

. Furthermore, we provide a general description of the transitive groups of degree p2 whose Sylow p-subgroup is isomorphic to

On non-Hamiltonian circulant digraphs of outdegree three

\mathbb{Z}_p \wr \mathbb{Z}_p

Products of similar matrices

, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p2, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p2, and find all nonnormal Cayley graphs of order p2.

New examples of compact Clifford-Klein forms of homogeneous spaces of SO(2, n)

SummaryLet Γ be a closed, cocompact subgroup of a simply connected, solvable Lie groupG, such that AdG Γ has the same Zariski closure as AdG. If α: Γ → GLn(ℝ) is any finite-dimensional representation of Γ, we show that α virtually extends to a representation ofG. (By combining this with work of Margulis on lattices in semisimple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that if Γ is isomorphic to a closed, cocompact subgroup Γ′ of another simply connected, solvable Lie groupG′, then any isomorphism from Γ to Γ′ extends to a crossed isomorphism fromG toG′. In the same vein, we prove a more concrete form of Mostows theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic.