Nilpotent right alternative rings and bol circle loops

We determine the nilpotent right alternative rings of prime power oirder pⁿ n ≥ 4, which are not left alternative. Those which are strongly right alternative become Bol loops under the circle operation. The smallest Bol circle loop has order 16. There are six such loops, all of which appear to be new.     

Albert's Construction for Semifields of Even Order

Albert's construction for commutative semifields of order 2 ⁿ , n odd, is presented. It avoids the construction of a presemifield and, in the case that n is prime, allows us to determine automorphism groups and the isomorphism classes. If n is a prime greater than three, the semifields are strictly not associative. These semifields are new for all n greater than three, differing from the binary semifields in that each admits only the trivial automorphism.

The authors present an explicit construction of an isotope of the 2⁵-element semifield that contains a subsemifield of order 2².     

On Restricted Leibniz Algebras

ABSTRACT

The role played by fields in relation to Galois Rings corresponds to semifields if the associativity is dropped, that is, if we consider Generalized Galois Rings instead of (associative) Galois rings. If S is a Galois ring and pS is the set of zero divisors in S, S* = S\ pS is known to be a finite {multiplicative} Abelian group that is cyclic if, and only if, S is a finite field, or S = ?/n? with n = 4 or n = p ^r for some odd prime p. Without associativity, S* is not a group, but a loop. The question of when this loop can be generated by a single element is addressed in this article.     

Values of the Euler Function in Various Sequences

Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n)^r = λ(n)^s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p − 1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1.     

Symmetry of iteration graphs

We examine iteration graphs of the squaring function on the rings ℤ/nℤ when n = 2 ^k p, for p a Fermat prime. We describe several invariants associated to these graphs and use them to prove that the graphs are not symmetric when k = 3 and when k ⩾ 5 and are symmetric when k = 4.     

The Primitive Idempotents of a Cyclic Group Algebra

Explicit expressions are obtained for the 2n + 1 primitive idempotents in FG, the semisimple group algebra of the cyclic group G of order pⁿ (p an odd prime, n ≥ 1) over the finite field F of prime power order q, when q has order φ(pⁿ)/2 modulo pⁿ.AMS Mathematical Subject Classification (2000): 20C05, 94B05, 12E20, 16S34.     

Presimplifiable and Cyclic Group Rings

Let R be any commutative ring with identity, and let C be a (finite or infinite) cyclic group. We show that the group ring R(C) is presimplifiable if and only if its augmentation ideal I(C) is presimplifiable. We conjecture that the group rings R(C _n ) are presimplifiable if and only if n = p ^m , p ∈ J(R), p is prime, and R is presimplifiable. We show the necessity of n = p ^m , and we prove the sufficiency when n = 2, 3, 4. These results were made possible by a new formula derived herein for the circulant determinantal coefficients.     

A survey of orthogonal arrays of strength two

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On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

On the matricial version of Fermat–Euler congruences

The congruences modulo the primary numbers n=p ^a are studied for the traces of the matrices A ⁿ and A ^n-φ(n), where A is an integer matrix and φ(n) is the number of residues modulo n, relatively prime to n. We present an algorithm to decide whether these congruences hold for all the integer matrices A, when the prime number p is fixed. The algorithm is explicitly applied for many values of p, and the congruences are thus proved, for instance, for all the primes p ≤ 7 (being untrue for the non-primary modulus n=6). We prove many auxiliary congruences and formulate many conjectures and problems, which can be used independently. Partially supported by RFBR, grant 05-01-00104. An erratum to this article is available at .     

The complexity and depth of Boolean circuits for multiplication and inversion in some fields <Emphasis Type="Italic">GF</Emphasis>(2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Let n = (p − 1) · p ^k , where p is a prime number such that 2 is a primitive root modulo p, and 2 ^p−1 − 1 is not a multiple of p ². For a standard basis of the field GF(2 ⁿ ), a multiplier of complexity O(log log p)n log n log log _p n and an inverter of complexity O(log p log log p)n log n log log _p n are constructed. In particular, in the case p = 3 the upper bound

Minimal Non—Commutative n—Insertive Rings

Let n be an integer with |n| > 1. If p is the smallest prime factor of |n|, we prove that a minimal non-commutative n-insertive ring contains n ⁴ elements and these rings have five (2p+4) isomorphic classes for p = 2 (p ≠ 2). This research is supported by the National Natural Science Foundation of China, and the Scientific Research Foundation for “Bai-Qian-Wan” Project, Fujian Province of China     

Frame starters and adders in dicyclic groups

We construct frame starters in dicyclic groups Q_2n, in particular we construct frame starters with adders in Q_2q, where q = pⁿ and p ≡ 3 mod 4 is a prime. We also deduce the existence of strong frame starters in Z_2n for odd integers n whose prime factors are congruent with 1 modulo 4. The obtained results imply the constructions of classes of Room frames of types 4^p and 2ⁿ. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 347–353, 1998     

Annihilators of power values of derivations in prime rings

In this paper we prove some results concerning annihilators of power values of derivations in prime rings. The following main theorem establishes a unified version of several earlier results in the literature:Let R be a prime ring with center Z and with extended centroid C,Q, its two-sided Martindale quotient ring, ρ a nonzero right ideal of R and D a nonzero derivation of R.Suppose that aD([x,y])ⁿ ∈ Z (D([x,y])ⁿa ∈ Z) for all x,y∈ρ where a∈Rand n is a fixed positive integer. If [ρ,ρ]ρ ≠ 0 and dim _C RC >4, then either aD(ρ) = 0 (a = 0 resp.) or D= ad(p) for some p∈ Qsuch that pρ = 0.     

Classes of Commutative Rings Defined by Special Conditions

ABSTRACT

In this article, we are mainly concerned with (n, d)-Krull rings, i.e., rings in which each n-presented prime ideal has height at most d. Precisely, we show that weakly n-Von Neumann regular rings are (n ? 1, 0)-Krull rings. Also, we prove that (n, d)-Krull property is not local property and that R is an (n, d)-Krull ring if and only if dim(R _P ) ≤ d for each n-presented prime ideal P of R. Finally, we construct a class of (2, d)-Krull rings which are neither (2, d ? 1)-Krull rings (for d = 1) nor (1, d)-Krull rings for d = 0,1.     

On an extension of a theorem on conjugacy class sizes

Let G be a finite group. We extend Alan Camina’s theorem on conjugacy classes sizes which asserts that if the conjugacy classes sizes of G are {1, p ^a , q ^b , p ^a q ^b }, where p and q are two distinct primes and a and b are integers, then G is nilpotent. We show that let G be a group and assume that the conjugacy classes sizes of elements of primary and biprimary orders of G are exactly {1, p ^a , n,p ^a n} with (p, n) = 1, where p is a prime and a and n are positive integers. If there is a p-element in G whose index is precisely p ^a , then G is nilpotent and n = q ^b for some prime q ≠ p.     

Cyclotomic numbers and primitive idempotents in the ring GF(q)[x]/(x−1)

Let q be an odd prime power and p be an odd prime with gcd(p,q)=1. Let order of q modulo p be f, and q^f=1+pλ. Here expressions for all the primitive idempotents in the ring R_pⁿ=GF(q)[x]/(x^pn−1), for any positive integer n, are obtained in terms of cyclotomic numbers, provided p does not divide λ if n2. The dimension, generating polynomials and minimum distances of minimal cyclic codes of length pⁿ over GF(q) are also discussed.     

Small quasimultiple affine and projective planes: Some improved bounds

We improve the known bounds on r(n): = min {λ| an (n², n, λ)-RBIBD exists} in the case where n + 1 is a prime power. In such a case r(n) is proved to be at most n + 1. If, in addition, n − 1 is the product of twin prime powers, then r(n) ${\ \le \ }{n \over 2}$. We also improve the known bounds on p(n): = min{λ| an (n² + n + 1, n + 1, λ)-BIBD exists} in the case where n² + n + 1 is a prime power. In such a case p(n) is bounded at worst by but better bounds could be obtained exploiting the multiplicative structure of GF(n² + n + 1). Finally, we present an unpublished construction by M. Greig giving a quasidouble affine plane of order n for every positive integer n such that n − 1 and n + 1 are prime powers. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 337–345, 1998     

Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.