Set Of Divisors Research Articles

On the composition factors of a group with the same prime graph as B n (5)

Let G be a finite group. The prime graph of G is a graph whose vertex set is the set of prime divisors of |G| and two distinct primes p and q are joined by an edge, whenever G contains an element of order pq. The prime graph of G is denoted by Γ(G). It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if G is a finite group such that Γ(G) = Γ(B n (5)), where n ⩾ 6, then G has a unique nonabelian composition factor isomorphic to B n (5) or C n (5).

The maximal energy of classes of integral circulant graphs

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count n and a set D of divisors of n in such a way that they have vertex set Zn and edge set {{a,b}:a,b∈Zn,gcd(a−b,n)∈D}. For a fixed prime power n=ps and a fixed divisor set size |D|=r, we analyse the maximal energy among all matching integral circulant graphs. Let pa1<pa2<⋯<par be the elements of D. It turns out that the differences di=ai+1−ai between the exponents of an energy maximal divisor set must satisfy certain balance conditions: (i) either all di equal q:=s−1r−1, or at most the two differences [q] and [q+1] may occur; (ii) there are rules governing the sequence d1,…,dr−1 of consecutive differences. For particular choices of s and r these conditions already guarantee maximal energy and its value can be computed explicitly.

Extremal energies of integral circulant graphs via multiplicativity

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. Integral circulant graphs can be characterised by their order n and a set D of positive divisors of n in such a way that they have vertex set Z/nZ and edge set {(a,b):a,b∈Z/nZ,gcd(a-b,n)∈D}. Among integral circulant graphs of fixed prime power order ps, those having minimal energy Eminps or maximal energy Emaxps, respectively, are known. We study the energy of integral circulant graphs of arbitrary order n with so-called multiplicative divisor sets. This leads to good bounds for Eminn and Emaxn as well as conjectures concerning the true value of Eminn.

Quadratic rational solvable groups

A finite group G is quadratic rational if all its irreducible characters are either rational or quadratic. If G is a quadratic rational solvable group, we show that the prime divisors of |G| lie in {2,3,5,7,13}, and no prime can be removed from this list. More generally, if G is solvable and the field Q(χ) generated by the values of χ over Q satisfies |Q(χ):Q|⩽k, for all χ∈Irr(G), then the set of prime divisors of |G| is bounded in terms of k. Also, we prove that the degree of the field generated by the values of all characters of a semi-rational solvable group (see Chillag and Dolfi, 2010 [1]) or a quadratic rational solvable group over Q is bounded, giving a positive answer to a question by D. Chillag and S. Dolfi.

On a problem of Erdős, Herzog and Schönheim

Let p1,p2,…,pn be distinct primes. In 1970, Erdős, Herzog and Schönheim proved that if D, |D|=m, is a set of divisors of N=p1α1⋯pnαn, α1≥α2≥⋯≥αn, no two members of the set being coprime and if no additional member may be included in D without contradicting this requirement then m≥αn∏i=1n−1(αi+1). They asked to determine all sets D such that the equality holds. In this paper we solve this problem. We also pose several open problems for further research.

On the relation between the non-commuting graph and the prime graph

‎$pi(G)$ denote the set of prime divisors of the order of $G$ and‎ ‎denote by $Z(G)$ the center of $G$‎. ‎Thetextit{ prime graph} of‎ ‎$G$ is the graph with‎ ‎vertex set $pi(G)$ where two distinct primes $p$ and $q$ are‎ ‎joined by an edge if and only if $G$‎ ‎contains an element of order $pq$ and the textit{non-commuting‎ ‎graph} of $G$ is the graph with the vertex set $G-Z(G)$ where two‎ ‎non-central elements $x$ and $y$ are‎ ‎joined by an edge if and only if $xy neq yx$‎. ‎Let $ G $ and $ H $ be non-abelian finite groups with isomorphic non-commuting graphs‎. ‎In this article‎, ‎we show that if $ | Z ( G ) | = | Z ( H ) | $‎, ‎then‎ ‎$ G $ and $ H $ have the same prime graphs and also‎, ‎the set of‎ ‎orders of the maximal abelian subgroups of $ G $ and $ H $ are‎ ‎the same‎.

ON THE MULTI-DIMENSIONAL PARTITIONS OF SMALL INTEGERS

For each dimension exceeds 1, determining the number of multi-dimensional partitions of a positive integer is an open question in combinatorial number theory. For n <TEX>${\leq}$</TEX> 14 and d <TEX>${\geq}$</TEX> 1 we derive a formula for the function <TEX>${\wp}_d(n)$</TEX> where <TEX>${\wp}_d(n)$</TEX> denotes the number of partitions of n arranged on a d-dimensional space. We also give an another definition of the d-dimensional partitions using the union of finite number of divisor sets of integers.

Deficiencies of holomorphic curves in algebraic varieties

We study property of Nevanlinna's deficiency as functions on linear systems in smooth complex projective algebraic varieties. We give a structure theorem for the set of deficient divisors. This structure theorem yields that the set of values of deficiency is at most countable. Moreover, we have a correspondence between the deficiencies and the linear systems.

Prescribed behavior of central simple algebras after scalar extension

1. Let A 1 , … , A n be central simple disjoint algebras over a field F. Let also l i | exp ( A i ) , m i | ind ( A i ) , l i | m i , and for each i = 1 , … , n , let l i and m i have the same sets of prime divisors. Then there exists a field extension E / F such that exp ( A i E ) = l i and ind ( A i E ) = m i , i = 1 , … , n . 2. Let A be a central simple algebra over a field K with an involution τ of the second kind. We prove that there exists a regular field extension E / K preserving indices of central simple K-algebras such that A ⊗ K E is cyclic and has an involution of the second kind extending τ.

On the norm of the nilpotent residuals of all subgroups of a finite group

R. Baer and Wielandt in 1934 and 1958, respectively, considered the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this paper, for a finite group G, we define the subgroup S ( G ) to be the intersection of the normalizers of the nilpotent residuals of all subgroups of G. Set S 0 = 1 . Define S i + 1 ( G ) / S i ( G ) = S ( G / S i ( G ) ) for i ⩾ 1 . By S ∞ ( G ) denote the terminal term of the ascending series. It is proved that G = S ∞ ( G ) if and only if the nilpotent residual G N is nilpotent. Furthermore, if all elements of prime order of G are in S ( G ) , then G is solvable and l p ( G ) ⩽ 1 , where l p ( G ) is p-length of G for p ∈ π ( G ) , where π ( G ) means the set of prime divisors of | G | .

Integral circulant graphs of prime power order with maximal energy

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count n and a set D of divisors of n in such a way that they have vertex set Z n and edge set { { a , b } : a , b ∈ Z n , gcd ( a - b , n ) ∈ D } . Using tools from convex optimization, we analyze the maximal energy among all integral circulant graphs of prime power order p s and varying divisor sets D . Our main result states that this maximal energy approximately lies between s ( p - 1 ) p s - 1 and twice this value. We construct suitable divisor sets for which the energy lies in this interval. We also characterize hyperenergetic integral circulant graphs of prime power order and exhibit an interesting topological property of their divisor sets.

On the Automorphism Group of Integral Circulant Graphs

The integral circulant graph $X_n (D)$ has the vertex set $Z_n = \{0, 1,\ldots$, $n{-}1\}$ and vertices $a$ and $b$ are adjacent, if and only if $\gcd(a{-}b$, $n)\in D$, where $D = \{d_1,d_2, \ldots, d_k\}$ is a set of divisors of $n$. These graphs play an important role in modeling quantum spin networks supporting the perfect state transfer and also have applications in chemical graph theory. In this paper, we deal with the automorphism group of integral circulant graphs and investigate a problem proposed in [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. We determine the size and the structure of the automorphism group of the unitary Cayley graph $X_n (1)$ and the disconnected graph $X_n (d)$. In addition, based on the generalized formula for the number of common neighbors and the wreath product, we completely characterize the automorphism groups $Aut (X_n (1, p))$ for $n$ being a square-free number and $p$ a prime dividing $n$, and $Aut (X_n (1, p^k))$ for $n$ being a prime power.

The energy of integral circulant graphs with prime power order

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b ? Zn; gcd(a-b, n)? D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

Further results on the perfect state transfer in integral circulant graphs

For a given graph G , denote by A its adjacency matrix and F ( t ) = exp ( i A t ) . We say that there exist a perfect state transfer (PST) in G if | F ( τ ) a b | = 1 , for some vertices a , b and a positive real number τ . Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant graphs (circulant graphs with integer eigenvalues). Some results on this topic have already been obtained by Saxena et al. (2007) [5] , Bašić et al. (2009) [6] and Basić & Petković (2009) [7] . In this paper, we show that there exists an integral circulant graph with n vertices having a perfect state transfer if and only if 4 ∣ n . Several classes of integral circulant graphs have been found that have a perfect state transfer for the values of n divisible by 4 . Moreover, we prove the nonexistence of a PST for several other classes of integral circulant graphs whose order is divisible by 4 . These classes cover the class of graphs where the divisor set contains exactly two elements. The obtained results partially answer the main question of which integral circulant graphs have a perfect state transfer.

On a group with three supersolvable subgroups of pairwise relatively prime indices

In this paper, we show that if G is a finite group with three supersolvable subgroups of pairwise relatively prime indices in G and G′ is nilpotent, then G is supersolvable. Let π(G) denote the set of prime divisors of |G| and max(π(G)) denote the largest prime divisor of |G|. We also establish that if G is a finite group such that G has three supersolvable subgroups H, K, and L whose indices in G are pairwise relatively prime, \({q \nmid p-1}\) where p = max(π(G)) and q = max(π(L)) with L a Hall p′-subgroup of G, then G is supersolvable.

The Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors

With a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincaré series of this filtration turns out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincaré series of the corresponding multi-index filtration similar to the one associated with plane germs. Here we show that the Poincaré series of a set of divisorial valuations on the ring of germs of functions of two variables defines “the topology of the set of the divisors” in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which shows that the Alexander polynomial is equivalent to the embedded topology.

Determination of the Biquaternion Divisors of Zero, Including the Idempotents and Nilpotents

The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the present paper. Using several different representations for biquaternions, we present simple formulae for the idempotents, nilpotents and divisors of zero, and we show that the complex components of a biquaternion divisor of zero must have a sum of squares that vanishes, and that this condition is equivalent to two conditions on the inner product of the real and imaginary parts of the biquaternion, and the equality of the norms of the real and imaginary parts. We give numerical examples of nilpotents, idempotents and other divisors of zero. Finally, we conclude with a statement about the composition of the set of biquaternion divisors of zero, and its subsets, the idempotents and the nilpotents.

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchywith complex-valued initial data and prove unique solvability globally intime for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inversealgebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.

On an analog of Pinkham's theorem for non-Tjurina components of rational singularities

There is a correspondence between the set of functions in the maximal ideal of the local ring of a rational surface singularity ξ and the set E + ( E ) consisting of certain effective divisors supported on the exceptional fiber E of a resolution of the singularity. Given an element Y ∈ E + ( E ) and a non-Tjurina component N of Y, we verify a formula for the least element of the set of divisors X ∈ E + ( E ) greater than or equal to Y + N stated but not proved in Tosun (1999). To cite this article: S. Altınok, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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 H K = K H . 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 ∩ 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 , … , p n , p n + 1 , … , p k } be the set of prime divisors of the order of a finite group G with { p 1 , … , p n } the set of prime divisors of the order of the normal subgroup N of G. A set of Sylow subgroups { P 1 , … , P n , P n + 1 , … , P k } , P i ∈ 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 ∈ { 1 , 2 , … , n } and j ∈ { 1 , 2 , … , 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.