Non-Cayley Vertex-Transitive Graphs of Order Twice the Product of Two Odd Primes

For a positive integer n, does there exist a vertex-transitive graph Γ on n vertices which is not a Cayley graph, or, equivalently, a graph Γ on n vertices such that Aut Γ is transitive on vertices but none of its subgroups are regular on vertices? Previous work (by Alspach and Parsons, Frucht, Graver and Watkins, Marusic and Scapellato, and McKay and the second author) has produced answers to this question if n is prime, or divisible by the square of some prime, or if n is the product of two distinct primes. In this paper we consider the simplest unresolved case for even integers, namely for integers of the form n = 2pq, where 2 < q < p, and p and q are primes. We give a new construction of an infinite family of vertex-transitive graphs on 2pq vertices which are not Cayley graphs in the case where p ≡ 1 (mod q). Further, if p ? 1 (mod q), p ≡ q ≡ 3(mod 4), and if every vertex-transitive graph of order pq is a Cayley graph, then it is shown that, either 2pq = 66, or every vertex-transitive graph of order 2pq admitting a transitive imprimitive group of automorphisms is a Cayley graph.     

Graphs on which a group of order <Emphasis Type="Italic">pq</Emphasis> acts edge-transitively

Let F be a finite simple undirected graph with no isolated vertices. Let p, q be prime numbers with p≥q. We complete the classification of the graphs on which a group of order pq acts edge-transitively. The results are the following. If Aut（Г） contains a subgroup G of order pq that acts edge-transitively on F, then F is one of the following graphs： （1） pK1,1; （2） pqK1,1; （3） pgq,1; （4） qKp,1 （p 〉 q）; （5） pCq （q 〉 2）; （6） qCp （p 〉 q）; （7） Cp （p 〉 q = 2）; （8） Cpq; （9） （Zp, C） whereC={±r^μ ｜μ∈Zq} withq〉2, q｜（p-1） and r≠1≡r^q （modp）; （10） Kp,1 （p 〉 q）; （11） a double Cayley graph B（G,C） with C = {1-r^μ ｜ μ ∈ Zq} and r≠1≡r^q （modp）; （12） Kpq,1;or （13） Kp,q.     

On a Criterion for Catalan's Conjecture

We give a new proof of a theorem of P. Mihailescu which states that the equation x ^p – y ^q = 1 is unsolvable with x, y integral and p, q odd primes, unless the congruences p ^q p (mod q ²) and q ^p q (mod p ²) hold.     

The old subvariety ofJ 0(pq) and the Eisenstein kernel in Jacobians

Whenp, q are distinct odd primes, and γ:J ₀(p)²×J ₀(q)²→J ₀(pq) is the natural map defined by the degeneracy maps, Ribet [10] determined the odd part of the kernel of γ. We study the 2-primary part of this kernel through its intersection with the Eisenstein kernelJ ₀(p)[I _p )²×J ₀(q)[I _q ]². We determine this intersection forp≢1 mod 16,q≢1 mod 16, and also produce new elements of ker γ wheneverp≡9 mod 16 orq≡9 mod 16. These sharpen Ribet's results in [10].     

More cyclic triplewhist tournaments

We extend the set of values of n for which it is known that a Z-cyclic triple whist tournament for 4n players exists by proving that if there exists such a tournament for q + 1 players, where q ≡ 3 (mod 4) is prime, then there exists such a tournament for qp^a1₁…p^an_n + 1 players, whenever the p_i are primes ≡ 5 (mod 8). © 1995 John Wiley & Sons, Inc.     

Existence of Z-cyclic 3PDTWh(<Emphasis Type="Italic">p</Emphasis>) for Prime <Emphasis Type="Italic">p</Emphasis> ≡ 1 (mod 4)

A directed triplewhist tournament on p players over Z _p is said to have the three-person property if no two games in the tournament have three common players. We briefly denote such a design as a 3PDTWh(p). In this paper, we investigate the existence of a Z-cyclic 3PDTWh(p) for any prime p ≡ 1 (mod 4) and show that such a design exists whenever p ≡ 5, 9, 13 (mod 16) and p ≥ 29. This result is obtained by applying Weil’s theorem. In addition, we also prove that a Z-cyclic 3PDTWh(p) exists whenever p ≡ 1 (mod 16) and p < 10, 000 except possibly for p = 257, 769. Gennian Ge’s Research was supported by National Natural Science Foundation of China under Grant No. 10471127, Zhejiang Provincial Natural Science Foundation of China under Grant No. R604001, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.     

Tables on fifth power residuacity

In this paper 2 ^p 1 (modq),q=10p+1,p 3 (mod 4),p andq prime, is expressed uniquely (except for changes in sign and interchange ofx, y) in the formq=w ²+25 (x ²+y ²)/2+125z ², 4wz=y ²–x ²–4xy, withw, x, y, z odd, forp<10⁵. For 10⁵<p<10⁶, allp such that 2 ^p 1 (mod 10p + 1),p 3 (mod 4),p and 10p + 1 prime, are listed.     

A Criterion for Elliptic Curves with Second Lowest 2-Power in L（1）（Ⅱ）

Let D = p1p2 …pm, where p1,p2, ……,pm are distinct rational primes with p1 ≡p2 ≡3（mod 8）, pi =1（mod 8）（3 ≤ i ≤ m）, and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the complex L-function of the congruent elliptic curve ED2 ： y^2 = x^3- D^2x at s = 1, divided by the period ω defined below, to be exactly divisible by 2^2m-2, the second lowest 2-power with respect to the number of the Gaussian prime factors of D. As a corollary, we obtain a new series of non-congruent numbers whose prime factors can be arbitrarily many. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer.     

On r*-sequenceability and symmetric harmoniousness of groups. II

In this paper we investigate symmetric harmoniousness of groups and connections of this concept to the R*-sequenceability of groups. We prove that, under suitable assumptions, the direct product of a symmetric harmonious group with a group that is R*-sequenceable is R*-sequenceable; we discuss the symmetric harmoniousness of abelian and of nilpotent groups; we also prove that, for a fixed odd prime p, all but possibly finitely many of the nonabelian groups of order pq (q prime, q ≡ 1 (mod p)) are symmetric harmonious. © 1995 John Wiley & Sons, Inc.     

On the prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis>) where <Emphasis Type="Italic">p</Emphasis> > 3 is a prime number

Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph. The third author was supported in part by a grant from IPM (No. 84200024).     

Some Multiply Derived Translation Planes with SL(2,5) as an Inherited Collineation Group in the Translation Complement

Finite translation planes having a collineation group isomorphic to SL(2,5) occur in many investigations on minimal normal non-solvable subgroups of linear translation complements. In this paper, we are looking for multiply derived translation planes of the desarguesian plane which have an inherited linear collineation group isomorphic to SL(2,5). The Hall plane and some of the planes discovered by Prohaska [10], see also [1], are translation planes of this kind of order q ²;, provided that q is odd and either q ²; 1 mod 5 or q is a power of 5. In this paper the case q ² -1 mod 5 is considered and some examples are constructed under the further hypothesis that either q 2 mod 3, or q 1 mod 3 and q 1 mod 4, or q -1 mod 4, 3 q and q 3,5 or 6 mod 7. One might expect that examples exist for each odd prime power q. But this is not always true according to Theorem 2.     

Some (17q, 17, 2) and (25q, 25, 3) BIBD Constructions

We give the explicit construction of a regular (17q, 17, 2)-BIBD for any prime power q 17 (mod 32) such that 2 is not a 4th power in GF(q) and the explicit construction of a regular (25q, 25, 3)-BIBD for any prime power q 25 (mod 48) such that and +3 are non-squares in GF(q).     

Class numbers of complex quadratic fields

Congruence conditions on the class numbers of complex quadratic fields have recently been studied by various investigators, including Barrucand and Cohn, Hasse, and the author. In this paper, we study the class number of Q(√ ? pq), where p ≡ q (mod 4) are distinct primes.     

Products of primes in conjugacy class sizes and irreducible character degrees

We prove that if the product pq of distinct primes p and q divides the degree of some irreducible complex character of a finite group G, then pq divides the size of some conjugacy class of G.     

Class number one quadratic fields and solvability of some Pellian equations

Consider these two types of positive square-free integers d≠ 1 for which the class number h of the quadratic field Q(√d) is odd: (1) d is prime∈ 1(mod 8), or d=2q where q is prime ≡ 3 (mod 4), or d=qr where q and r are primes such that q≡ 3 (mod 8) and r≡ 7 (mod 8); (2) d is prime ≡ 1 (mod 8), or d=qr where q and r are primes such that q≡r≡ 3 or 7 (mod 8). For d of type (2) (resp. (1)), let Π be the set of all primes (resp. odd primes) p∈N satisfying (d/p) = 1. Also, let δ :=0 (resp. δ :=1) if d≡ 2,3 (mod 4) (resp. d≡ 1 (mod 4)). Then the following are equivalent: (a) h=1; (b) For every p∈П at least one of the two Pellian equations Z ²-dY ² = ±4^δ p is solvable in integers. (c) For every p∈П the Pellian equation W ²-dV ² = 4^δ p ² has a solution (w,v) in integers such that gcd (w,v) divides 2^δ.     

The Orders of Vertices of a Character-degree Graph

Let G be a finite group. We put ρ(G) = {p|p is a prime dividing χ(1) for some χ ∈Irr(G)}. We define a graph Γ(G), whose vertices are the primes in ρ(G) and p, q ∈ ρ(G) are connected in Γ(G) denoted p ～ q, if pq||χ(1) for some χ ∈Irr(G). For p ∈ ρ(G), we define ord(p) = |{q ∈ ρ(G)|q ～ p}|. We call ord(p) the order of the vertex p of the graph Γ(G). In this article, we discuss orders and the influences of orders on the structure of finite groups.     

Some Results Connected with the Class Number Problem in Real Quadratic Fields

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.     

Mordell-Weil groups and Selmer groups of twin- prime elliptic curves

Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.     

Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

On runs of consecutive quadratic residues and quadratic nonresidues

Based on results of Weil and of Burgess, we have obtained a boundK(l) such that all primesp K(l) have a sequence of at leastl consecutive quadratic residues and a sequence of at leastl consecutive nonresidues in the interval [1,p – 1]. The bound forl=9 being 414463, we have computed, for primes less than 420000, the lengths of the longest sequences of consecutive residues and of nonresidues. We present these data and make some observations concerning them. One of the observations is that there is an observed difference in the length of the maximal sequence between primes congruent to 1 (mod 4) and primes congruent to 3 (mod 4).