We find a relationship between regular embeddings of G, an elementary abelian p-group of order p, into unipotent upper triangular matrices with entries in Fp and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n = 3 up to linear isomorphism, and use that classification to determine the number of Hopf Galoi...

0.1. Let A5 be the group defined by generators x2, x3, x5 and relations x 2 2 = x 3 3 = x5 = x2x3x5 = 1. It is well known that A5 is isomorphic to the alternating group in 5 letters. Let k be an algebraically closed field of characteristic p where p = 0 or a prime ≥ 7. Let G be a connected reductive algebraic group over k. We shall be interested in the classification of homomorphisms A5 −→ G up...

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v1, . . . , vk of G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graph...

We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P (x) with L regular over any PAC field P of characteristic zero. A special case of this implies that G is a Galois group over Fp(x) for almost all primes p.

A 1-factorisation of a graph G is a partition of the edge set of G into 1factors (perfect matchings); a perfect 1-factorisation of G is a 1-factorisation of G in which the union of any two of the 1-factors is a Hamilton cycle in G. It is known that for bipartite 4-regular circulant graphs, having order 2 (mod 4) is a necessary (but not sufficient) condition for the existence of a perfect 1-fact...

Hilton, A. J. W., Alternating Hamiltonian circuits in edge-coloured bipartite graphs, Discrete Applied Mathematics 35 (1992) 271-273. We show that if G = K,, is edge-coloured with t-12 colours so that the subgraph induced by the edges of each colour is regular of order 2n, then G has a Hamiltonian circuit in which adjacent edges have different colours. We also give a generalization of this resu...

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. We show that a symmetric cubic graph of girth at most 9 is either 1-regular or 2-regular (following the notation of Djokovic), or...

A regular covering projection ℘: X̃ → X of connected graphs is G-admissible if G lifts along ℘. Denote by G̃ the lifted group, and let CT(℘) be the group of covering transformations. The projection is called G-split whenever the extension CT(℘) → G̃ → G splits. In this paper, split 2-covers are considered, with a particular emphasis given to cubic symmetric graphs. Supposing that G is transitive o...

We define regular Kac-Moody superalgebras and classify them using integrable modules. We give conditions for irreducible highest weight modules of regular Kac-Moody superalgebras to be integrable. This paper is a major part of the proof for the classification of finite-growth contragredient Lie superalgebras. The results of this paper are a crucial part of the proof for the classification of co...

It was conjectured in [Nam73] that the Torelli map Mg → Ag associating to a curve its jacobian extends to a regular map from the DeligneMumford moduli space of stable curves Mg to the (normalization of the) Igusa blowup A cent g . A counterexample in genus g = 9 was found in [AB11]. Here, we prove that the extended map is regular for all g ≤ 8, thus completely solving the problem in every genus.