Holomorphic Cartan geometries on complex tori

In [6], it was asked whether all flat holomorphic Cartan geometries $(G,H)$ on a complex torus are translation invariant. We answer this affirmatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type $(G,H)$, with G a complex affine Lie group, on any complex torus is translation invariant.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

Résolutions flasques des groupes réductifs connexes

A connected reductive group G over a (characteristic zero) field k may be written as a quotient $H/S$, where the k-group H is an extension of a quasitrivial torus by a simply connected group, and S is a flasque k-torus, central in H. Such presentations $G=H/S$ lead to a simplified approach to the Galois cohomology of G and related objects, such as the Brauer group of a smooth compactification of G. When k is a number field, one also recovers known formulas, in terms of S, for the quotient of the group $G(k)$ of rational points by R-equivalence, and for the Abelian groups which measure the lack of weak approximation and the failure of the Hasse principle for principal homogeneous spaces. To cite this article: J.-L. Colliot-Thélène, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Forwarding and optical indices of 4-regular circulant networks

An all-to-all routing in a graph G is a set of oriented paths of G, with exactly one path for each ordered pair of vertices. The load of an edge under an all-to-all routing R is the number of times it is used (in either direction) by paths of R, and the maximum load of an edge is denoted by $\pi (G,R)$. The edge-forwarding index $\pi (G)$ is the minimum of $\pi (G,R)$ over all possible all-to-all routings R, and the arc-forwarding index $\overrightarrow{\pi }(G)$ is defined similarly by taking direction into consideration, where an arc is an ordered pair of adjacent vertices. Denote by $w(G,R)$ the minimum number of colours required to colour the paths of R such that any two paths having an edge in common receive distinct colours. The optical index $w(G)$ is defined to be the minimum of $w(G,R)$ over all possible R, and the directed optical index $\overrightarrow{w}(G)$ is defined similarly by requiring that any two paths having an arc in common receive distinct colours. In this paper we obtain lower and upper bounds on these four invariants for 4-regular circulant graphs with connection set $\{\pm 1,\pm s\}$, $1<s<n/2$. We give approximation algorithms with performance ratio a small constant for the corresponding forwarding index and routing and wavelength assignment problems for some families of 4-regular circulant graphs.     

On Fano and weak Fano Bott–Samelson–Demazure–Hansen varieties

Let G be a simple algebraic group over the field of complex numbers. Fix a maximal torus T and a Borel subgroup B of G containing T. Let w be an element of the Weyl group W of G, and let $Z(\stackrel{?}{w})$ be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression $\stackrel{?}{w}$ of w with respect to the data $(G,B,T)$.In this article we give complete characterization of the expressions $\stackrel{?}{w}$ such that the corresponding BSDH variety $Z(\stackrel{?}{w})$ is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results.     

Closed walks and eigenvalues of Abelian Cayley graphs

We show that Abelian Cayley graphs contain many closed walks of even length. This implies that given $k?3$, for each $\mathrm{?}>0$, there exists $C=C(\mathrm{?},k)>0$ such that for each Abelian group G and each symmetric subset S of G with $1?S$, the number of eigenvalues ${\lambda }_i$ of the Cayley graph $X=X(G,S)$ such that ${\lambda }_i?k?\mathrm{?}$ is at least $C?|G|$. This can be regarded as an analogue for Abelian Cayley graphs of a theorem of Serre for regular graphs. To cite this article: S.M. Cioab?, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Solvable PST-groups,strong Sylow bases and mutually permutable products

A subgroup H of a group G is said to permute with the subgroup K of G if $HK=KH$. Subgroups H and K are mutually permutable (totally permutable) in G if every subgroup of H permutes with K and every subgroup of K permutes with H (if every subgroup of H permutes with every subgroup of K). If H and K are mutually permutable and $H\cap K=1$, then H and K are totally permutable. A subgroup H of G is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a PST-group if S-permutability is a transitive relation in G. Let $\{p_1,\dots ,p_n,p_{n+1},\dots ,p_k\}$ be the set of prime divisors of the order of a finite group G with $\{p_1,\dots ,p_n\}$ the set of prime divisors of the order of the normal subgroup N of G. A set of Sylow subgroups $\{P_1,\dots ,P_n,P_{n+1},\dots ,P_k\}$, $P_i\in {\mathrm{Syl}}_{p_i}(G)$, form a strong Sylow system with respect to N if $P_i{P}_j$ is a mutually permutable product for all $i\in \{1,2,\dots ,n\}$ and $j\in \{1,2,\dots ,k\}$. We show that a finite group G is a solvable PST-group if and only if it has a normal subgroup N such that $G/N$ is nilpotent and G has a strong Sylow system with respect to N. It is also shown that G is a solvable PST-group if and only if G has a normal solvable PST-subgroup N and $G/N^{″}$ is a solvable PST-group.     

An exact upper bound for sums of element orders in non-cyclic finite groups

Denote the sum of element orders in a finite group G by $\psi (G)$ and let $C_n$ denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that $\psi (G)\le \frac{7}{11}\psi (C_n)$ and $\psi (G)<\frac{1}{q?1}\psi (C_n)$. The first result is best possible, since for each $n=4k$, k odd, there exists a group G of order n satisfying $\psi (G)=\frac{7}{11}\psi (C_n)$ and the second result implies that if G is of odd order, then $\psi (G)<\frac{1}{2}\psi (C_n)$. Our results improve the inequality $\psi (G)<\psi (C_n)$ obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some $\psi (G)$-based sufficient conditions for the solvability of G.     

Histories in Path Graphs

For a given graph G and a positive integer r the r-path graph, $P_r(G)$, has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length $r-1$, and their union forms either a cycle or a path of length $r+1$ in G. Let $P_r^k(G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_r^k(G)$. The k-history $P_r^{-k}(H)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of H. We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.     

Rank,term rank and chromatic number of a graph

Let G be a graph with a nonempty edge set, we denote the rank of the adjacency matrix of G and the term rank of G, by $\mathrm{rk}(G)$ and $\mathrm{Rk}(G)$, respectively. It was conjectured [C. van Nuffelen, Amer. Math. Monthly 83 (1976) 265–266], for any graph G, $\chi (G)?\mathrm{rk}(G)$. The first counterexample to this conjecture was obtained by Alon and Seymour [J. Graph Theor. 13 (1989) 523–525]. Recently, Fishkind and Kotlov [Discrete Math. 250 (2002) 253–257] have proved that for any graph G, $\chi (G)?\mathrm{Rk}(G)$. In this Note we improve Fishkind–Kotlov upper bound and show that $\chi (G)?\frac{\mathrm{rk}(G)+\mathrm{Rk}(G)}{2}$. To cite this article: S. Akbari, H.-R. Fanaï, C. R. Acad. Sci. Paris, Ser. I 340 (2005).