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.     

Representations of polynomial Rota–Baxter algebras

A Rota–Baxter operator is an algebraic abstraction of integration, which is the typical example of a weight zero Rota–Baxter operator. We show that studying the modules over the polynomial Rota–Baxter algebra $(\mathbf{k}[x],P)$ is equivalent to studying the modules over the Jordan plane, and we generalize the direct decomposability results for the $(\mathbf{k}[x],P)$-modules in [13] from algebraically closed fields of characteristic zero to fields of characteristic zero. Furthermore, we provide a classification of Rota–Baxter modules up to isomorphism based on indecomposable $\mathbf{k}[x]$-modules.     

Partial cohomology of groups and extensions of semilattices of abelian groups

We extend the notion of a partial cohomology group $H^n(G,A)$ to the case of non-unital A and find interpretations of $H^1(G,A)$ and $H^2(G,A)$ in the theory of extensions of semilattices of abelian groups by groups.     

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.     

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.     

Primeness property for graded central polynomials of verbally prime algebras

Let F be an infinite field. The primeness property for central polynomials of $M_n(F)$ was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for $M_n(F)$ and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider $M_n(R)$, where R admits a regular grading, with a grading such that $M_n(F)$ is a homogeneous subalgebra and provide sufficient conditions – satisfied by $M_n(E)$ with the trivial grading – to prove that $M_n(R)$ has the primeness property if $M_n(F)$ does. We also prove that the algebras $M_{a,b}(E)$ satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property.     

Annihilator-stability and unique generation

A ring R is said to be left uniquely generated if $Ra=Rb$ in R implies that $a=ub$ for some unit u in R. These rings have been of interest since Kaplansky introduced them in 1949 in his classic study of elementary divisors. Writing $\mathtt{l}(b)=\{r\in R|rb=0\}$, a theorem of Canfell asserts that R is left uniquely generated if and only if, whenever $Ra+\mathtt{l}(b)=R$ where $a,b\in R$, then $a?u\in \mathtt{l}(b)$ for some unit u in R. By analogy with the stable range 1 condition we call a ring with this property left annihilator-stable. In this paper we exploit this perspective on the left UG rings to construct new examples and derive new results. For example, writing J for the Jacobson radical, we show that a semiregular ring R is left annihilator-stable if and only if $R/J$ is unit-regular, an analogue of Bass' theorem that semilocal rings have stable range 1.     

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).     

Quasi-Baer module hulls and applications

Let V be a module with $S=\mathrm{End}(V)$. V is called a quasi-Baer module if for each ideal J of S, $r_V(J)=eV$ for some $e^2=e\in S$. On the other hand, V is called a Rickart module if for each $?\in S$, $\mathrm{Ker}(?)=eV$ for some $e^2=e\in S$. For a module N, the quasi-Baer module hull $\mathbf{qB}(N)$ (resp., the Rickart module hull $\mathbf{Ric}(N)$) of N, if it exists, is the smallest quasi-Baer (resp., Rickart) overmodule, in a fixed injective hull $E(N)$ of N. In this paper, we initiate the study of quasi-Baer and Rickart module hulls. When a ring R is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective R-module has a quasi-Baer hull. Let R be a Dedekind domain with F its field of fractions and let $\{K_i|i\in \mathrm{\Lambda }\}$ be any set of R-submodules of $F_R$. For an R-module $M_R$ with ${\mathrm{Ann}}_R(M)\ne 0$, we show that $M_R\oplus {({?}_{i\in \mathrm{\Lambda }}{K}_i)}_R$ has a quasi-Baer module hull if and only if $M_R$ is semisimple. This quasi-Baer hull is explicitly described. An example such that $M_R\oplus {({?}_{i\in \mathrm{\Lambda }}{K}_i)}_R$ has no Rickart module hull is constructed. If N is a module over a Dedekind domain for which $N/t(N)$ is projective and ${\mathrm{Ann}}_R(t(N))\ne 0$, where $t(N)$ is the torsion submodule of N, we show that the quasi-Baer hull $\mathbf{qB}(N)$ of N exists if and only if $t(N)$ is semisimple. We prove that the Rickart module hull also exists for such modules N. Furthermore, we provide explicit constructions of $\mathbf{qB}(N)$ and $\mathbf{Ric}(N)$ and show that in this situation these two hulls coincide. Among applications, it is shown that if N is a finitely generated module over a Dedekind domain, then N is quasi-Baer if and only if N is Rickart if and only if N is Baer if and only if N is semisimple or torsion-free. For a direct sum $N_R$ of finitely generated modules, where R is a Dedekind domain, we show that N is quasi-Baer if and only if N is Rickart if and only if N is semisimple or torsion-free. Examples exhibiting differences between the notions of a Baer hull, a quasi-Baer hull, and a Rickart hull of a module are presented. Various explicit examples illustrating our results are constructed.     

Sur le groupe affine d'un corps local

Let G be the affine group of a local field and $A(G)$ be its Fourier algebra. We prove some properties for G of which the principal is the following: if G is Archimedian and $f\in A(G)$ with compact support for which the Fourier cotransform is of finite rank, then $f=0$. To cite this article: W. Nasserddine, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Some extensions of the Cauchy-Davenport theorem

The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s in $\mathbb{Z}/p\mathbb{Z}$, the cardinality of the sumset $A+B=\{a+b|a\in A,b\in B\}$ is bounded below by $\min (r+s-1,p)$; moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G and positive integers $r,s⩽|G|$, the analogous sharp lower bound, namely the function${\mu }_G(r,s)=\min \{|A+B||A,B\subset G,|A|=r,|B|=s\}.$ Important progress on this topic has been achieved in recent years, leading to the determination of ${\mu }_G$ for all abelian groups G. In this note we survey the history of earlier results and the current knowledge on this function.     

First order theory of cyclically ordered groups

By a result known as Rieger's theorem (1956), there is a one-to-one correspondence, assigning to each cyclically ordered group H a pair $(G,z)$ where G is a totally ordered group and z is an element in the center of G, generating a cofinal subgroup $〈z〉$ of G, and such that the cyclically ordered quotient group $G/〈z〉$ is isomorphic to H. We first establish that, in this correspondence, the first-order theory of the cyclically ordered group H is uniquely determined by the first-order theory of the pair $(G,z)$. Then we prove that the class of cyclically orderable groups is an elementary class and give an axiom system for it. Finally we show that, in contrast to the fact that all theories of totally ordered Abelian groups have the same universal part, there are uncountably many universal theories of Abelian cyclically ordered groups.