The tame-wild principle for discriminant relations for number fields

Consider tuples of separable algebras over a common local or global number field, related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.Comment: 31 pages, 11 figures. Version 2 fixes a normalization error: |G| is corrected to n in Section 7.5. Version 3 fixes an off-by-one error in Section 6.

On quasi-orthogonal cocycles

We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square {±1}-matrices of size congruent to 2 modulo 4. Quasi-orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi-Hadamard groups, relative quasi-difference sets, and certain partially balanced incomplete block designs, are proved.Junta de Andalucía FQM-01

On 2-arc-transitivity of Cayley graphs

AbstractThe classification of 2-arc-transitive Cayley graphs of cyclic groups, given in (J. Algebra. Combin. 5 (1996) 83–86) by Alspach, Conder, Xu and the author, motivates the main theme of this article: the study of 2-arc-transitive Cayley graphs of dihedral groups. First, a previously unknown infinite family of such graphs, arising as covers of certain complete graphs, is presented, leading to an interesting property of Singer cycles in the group PGL(2,q), q an odd prime power, among others. Second, a structural reduction theorem for 2-arc-transitive Cayley graphs of dihedral groups is proved, putting us—modulo a possible existence of such graphs among regular cyclic covers over a small family of certain bipartite graphs—a step away from a complete classification of such graphs. As a byproduct, a partial description of 2-arc-transitive Cayley graphs of abelian groups with at most three involutions is also obtained

The Tame-Wild Principle for Discriminant Relations for Number Fields

Consider tuples ( K1 , … , Kr ) of separable algebras over a common local or global number field F1, with the Ki related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants of Ki ∕ F . We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification

On the split structure of lifted groups

Let ▫$wp colon tilde{X} to X$▫ be a regular covering projection of connected graphs with the group of covering transformations ▫$rm{CT}_wp$▫ being abelian. Assuming that a group of automorphisms ▫$G le rm{Aut} X$▫ lifts along $wp$ to a group ▫$tilde{G} le rm{Aut} tilde{X}$▫, the problem whether the corresponding exact sequence ▫$rm{id} to rm{CT}_wp to tilde{G} to G to rm{id}$▫ splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence. In the above combinatorial setting the extension is given only implicitly: neither ▫$tilde{G}$▫ nor the action ▫$Gto rm{Aut} rm{CT}_wp$▫ nor a 2-cocycle ▫$G times G to rm{CT}_wp$▫, are given. Explicitly constructing the cover ▫$tilde{X}$▫ together with ▫$rm{CT}_wp$▫ and ▫$tilde{G}$▫ as permutation groups on ▫$tilde{X}$▫ is time and space consuming whenever ▫$rm{CT}_wp$▫ is largethus, using the implemented algorithms (for instance, HasComplement in Magma) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group)one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever ▫$rm{CT}_wp$▫ is elementary abelian.Naj bo ▫$wp colon tilde{X} to X$▫ regularna krovna projekcija povezanih grafov, grupa krovnih transformacij ▫$rm{CT}_wp$▫ pa naj bo abelova. Ob predpostavki, da se grupa avtomorfizmov ▫$G le rm{Aut} X$▫ dvigne vzdolž ▫$wp$▫ do grupe ▫$tilde{G} le rm{Aut} tilde{X}$▫, podrobno analiziramo problem, ali se ustrezno eksaktno zaporedje ▫$rm{id} to rm{CT}_wp to tilde{G} to G to rm{id}$▫ razcepi glede na Cayleyevo dodelitev napetosti, ki rekonstruira projekcijo do ekvivalence natančno. V gornjem kombinatoričnem sestavu je razširitev podana samo implicitno: podani niso ne ▫$tilde{G}$▫ ne delovanje ▫$Gto rm{Aut} rm{CT}_wp$▫ ne 2-kocikel ▫$G times G to rm{CT}_wp$▫. Eksplicitno konstruiranje krova ▫$tilde{X}$▫ ter ▫$rm{CT}_wp$▫ in ▫$tilde{G}$▫ kot permutacijskih grup na ▫$tilde{X}$▫ je časovno in prostorsko zahtevno vselej, kadar je ▫$rm{CT}_wp$▫ veliktako je uporaba implementiranih algoritmov (na primer, HasComplement v Magmi) vse prej kot optimalna. Namesto tega pokažemo, da lahko najnujnejšo informacijo o delovanju in 2-kociklu učinkovito izluščimo neposredno iz napetosti (ne da bi eksplicitno konstruirali krov in dvignjeno grupo)zdaj bi bilo mogoče uporabiti standardno metodo reduciranja problema na reševanje sistema linearnih enačb nad celimi števili. Vendar tukaj uberemo malce drugačen pristop, ki sploh ne zahteva nobenega poznavanja kohomologije. Časovno in prostorsko zahtevnost formalno analiziramo za vse primere, ko je ▫$rm{CT}_wp$▫ elementarna abelova