Further Rigid Triples of Classes in G2G_{2}

We establish the existence of two rigid triples of conjugacy classes in the algebraic group $G_{2}$ in characteristic $5$, complementing results of the second author with Liebeck and Marion. As a corollary, the finite groups $G_{2}(5^n)$ are not $(2,4,5)$-generated, confirming a conjecture of Marion in this case.Comment: 5 pages. To appear in International Journal of Group Theor

Graham Higman's lectures on januarials

This is an account of a series of lectures of Graham Higman on "januarials", namely coset graphs for actions of triangle groups which become 2-face maps when embedded in orientable surfaces.Comment: 22 pages, 13 figure

Tight orientably-regular polytopes

Every equivelar abstract polytope of type $\{p_1, \ldots, p_{n-1}\}$ has at least $2p_1 \cdots p_{n-1}$ flags. Polytopes that attain this lower bound are called tight. Here we investigate the question of under what conditions there is a tight orientably-regular polytope of type $\{p_1, \ldots, p_{n-1}\}$. We show that it is necessary and sufficient that whenever $p_i$ is odd, both $p_{i-1}$ and $p_{i+1}$ are even divisors of $2p_i$.Comment: 15 page

Skew product groups for monolithic groups

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free in $G$. In this paper, we classify all examples in which $B$ is monolithic (meaning that it has a unique minimal normal subgroup, and that subgroup is not abelian) and core-free in $G$. As a consequence, we obtain a classification of all proper skew morphisms of finite non-abelian simple groups

Vertex-transitive Haar graphs that are not Cayley graphs

In a recent paper (arXiv:1505.01475 ) Est\'elyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph $G(10,2)$, occurring as the graph $40$ in the Foster census of connected symmetric trivalent graphs.Comment: 9 pages, 2 figure

Half-arc-transitive graphs of arbitrary even valency greater than 2

A half-arc-transitive graph is a regular graph that is both vertex- and edge-transitive, but is not arc-transitive. If such a graph has finite valency, then its valency is even, and greater than $2$. In 1970, Bouwer proved that there exists a half-arc-transitive graph of every even valency greater than 2, by giving a construction for a family of graphs now known as $B(k,m,n)$, defined for every triple $(k,m,n)$ of integers greater than $1$ with $2^m \equiv 1 \mod n$. In each case, $B(k,m,n)$ is a $2k$-valent vertex- and edge-transitive graph of order $mn^{k-1}$, and Bouwer showed that $B(k,6,9)$ is half-arc-transitive for all $k > 1$. For almost 45 years the question of exactly which of Bouwer's graphs are half-arc-transitive and which are arc-transitive has remained open, despite many attempts to answer it. In this paper, we use a cycle-counting argument to prove that almost all of the graphs constructed by Bouwer are half-arc-transitive. In fact, we prove that $B(k,m,n)$ is arc-transitive only when $n = 3$, or $(k,n) = (2,5)$, % and $m$ is a multiple of $4$, or $(k,m,n) = (2,3,7)$ or $(2,6,7)$ or $(2,6,21)$. In particular, $B(k,m,n)$ is half-arc-transitive whenever $m > 6$ and $n > 5$. This gives an easy way to prove that there are infinitely many half-arc-transitive graphs of each even valency $2k > 2$.Comment: 16 pages, 1 figur

Magnetothermoelectric effects in Fe{1+d}Te{1-x}Se{x}

We report resistivity as well as the Hall, Seebeck and Nernst coefficients data for Fe{1+d}Te{1-x}Se{x} single crystals with x = 0, 0.38, and 0.40. In the parent compound Fe{1.04}Te we observe at Tn = 61 K a sudden change of all quantities studied, which can be ascribed to the Fermi surface reconstruction due to onset of the antiferromagnetic order. Two very closely doped samples: Fe{1.01}Te{0.62}Se{0.38} (Se38) and Fe{1.01}Te{0.60}Se{0.40} (Se40) are superconductors with Tc = 13.4 K and 13.9 K, respectively. There are no evident magnetic transitions in either Se38 or Se40. Properties of these two single crystals are almost identical at high temperatures, but start to diverge below T ~ 80 K. Perhaps we see the onset of scattering that might be a related to changes in short range magnetic correlations caused by selenium doping.Comment: 14 pages, 4 figure

Relevance of electron-lattice coupling in cuprates superconductors

We study the oxygen isotope (^{16}O,^{18}O) and finite size effects in Y_{1-x}Pr_{x}Ba_{2}Cu_{3}O_{7-\delta} by in-plane penetration depth (\lambda _{ab}) measurements. A significant change of the length L_{c} of the superconducting domains along the c-axis and \lambda_{ab}^{2} is deduced, yielding the relative isotope shift \Delta L_{c}/L_{c}\approx \Delta \lambda _{ab}^{2}/\lambda_{ab}^{2}\approx -0.14 for x=0, 0.2 and 0.3. This uncovers the existence and relevance of the coupling between the superfluid, lattice distortions and anharmonic phonons which involve the oxygen lattice degrees of freedom.Comment: 4 pages, 3 figure

On the orders of arc-transitive graphs

A graph is called {\em arc-transitive} (or {\em symmetric}) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer $k$, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is $kp$ for some prime $p$, and that if $d\ge 4$, then there exist only finitely many connected $d$-valent 2-arc-transitive graphs whose order is $kp$ or $kp^2$ for some prime $p$. We also prove that there are infinitely many (even) values of $k$ for which there are only finitely many connected 3-valent symmetric graphs of order $kp$ where $p$ is prime

Strong Kochen-Specker theorem and incomputability of quantum randomness

The Kochen-Specker theorem shows the impossibility for a hidden variable theory to consistently assign values to certain (finite) sets of observables in a way that is non-contextual and consistent with quantum mechanics. If we require non-contextuality, the consequence is that many observables must not have pre-existing definite values. However, the Kochen-Specker theorem does not allow one to determine which observables must be value indefinite. In this paper we present an improvement on the Kochen-Specker theorem which allows one to actually locate observables which are provably value indefinite. Various technical and subtle aspects relating to this formal proof and its connection to quantum mechanics are discussed. This result is then utilized for the proposal and certification of a dichotomic quantum random number generator operating in a three-dimensional Hilbert space.Comment: 31 pages, 5 figures, final versio