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Abstract: Let K be a perfect field of characteristic not equal to two, an algebraic closure of K and let be the Galois group of the extension . Let T be an n-point set in . The field of moduli of T is contained in each field of definition but it is not necessarily a field of definition. In this paper we show that point sets of odd cardinality in with field of moduli K are defined over their field of moduli. We, also, show that, except for the special case of the 4-point sets, this does not hold in general for point sets of even cardinality . Finally we prove that the following local-to-global principle holds for point sets of cardinality : if an n-point set T in is defined over for each prime p (including the prime at ∞), then it is defined over . From this result we derive an analogue local-to-global principle for hyperelliptic curves. [Copyright &y& Elsevier]

Abstract: Let p be an odd prime number. For the cyclotomic -extension of a finite algebraic number field F, we denote by the maximal unramified pro-p-extension of . In this paper, using Iwasawa theory and the theory of central class fields, we give necessary conditions for to be abelian, and give sufficient conditions in certain special cases. [Copyright &y& Elsevier]

Abstract: The number , appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many partial results about the minimal polynomial of this algebraic number. Here we obtain the general formula and it is Möbius inversion for this minimal polynomial by means of the Dickson polynomials and the Möbius inversion theory. Moreover, we investigate the homogeneous cyclotomic, Chebychev and Dickson polynomials in two variables and we show that our main results in one variable case nicely extend to this situation. In this paper, the deep results concerning these polynomials are proved by elementary arguments. [Copyright &y& Elsevier]

We study a $ \mathbb{Z}G $-module A such that $ \mathbb{Z} $ is the ring of integer numbers, the group G has an infinite sectional p-rank (or an infinite 0-rank), C( A) = 1, A is not a minimax $ \mathbb{Z} $-module, and, for any proper subgroup H of infinite sectional p-rank (or infinite 0-rank, respectively), the quotient module A/ C( H) is a minimax $ \mathbb{Z} $-module. It is shown that if the group G is locally soluble, then it is soluble. Some properties of soluble groups of this kind are discussed. [ABSTRACT FROM AUTHOR]

Abstract: The family of semiabelian p-groups is the minimal family that contains {1} and is closed under quotients and semidirect products with finite abelian p-groups. Kisilevsky and Sonn have solved the minimal ramification problem for a certain subfamily of the family of semiabelian p-groups. We show that is in fact the entire family of semiabelian p-groups and by this complete their solution to all semiabelian p-groups. [Copyright &y& Elsevier]

Abstract: This paper gives the answer to a question of R.M. Wilson regarding the existence of group divisible designs of large order. Let k and u be positive integers such that . Then there exists an integer such that there exists a group divisible design of group type with block size k and index one for any integer satisfying the necessary arithmetic conditions [1.] , [2.] . This paper also presents a large-index asymptotic existence theorem for group divisible t-designs with a fixed number of groups, fixed group size and fixed block size. [Copyright &y& Elsevier]

Abstract: For any odd prime , we present a reflection theorem which for is the Scholz Reflection Theorem. We obtain some formulas for the -rank and the -rank(), where is a number field. With the help of our reflection theorem, we establish relations between the -divisibility of the order of and the -divisibility of the class number of some algebraic number fields . [Copyright &y& Elsevier]

Abstract: We give the values of the irreducible complex characters of the group where ℓ is an integer >1. [Copyright &y& Elsevier]

In a paper of 2003, Külshammer, Olsson and Robinson defined ℓ-blocks for the symmetric groups, where ℓ>1 is an arbitrary integer. In this paper, we give a definition for the defect group of the principal ℓ-block. We then check that, in the Abelian case, we have an analogue of one of Broué's conjectures. [ABSTRACT FROM PUBLISHER]

A special matrix semigroup acting on the set of primitive pairs of positive integers naturally parameterizing certainDiophantine equations (such as those defining Pythagorean triples) is constructed and studied. The Pythagorean triples are related to a trichotomy tree of solutions. [ABSTRACT FROM AUTHOR]

The article clarifies and describes the algebraic structure of H• (n; C). The structure is defined as the structure of a supercommulative superalgebra in the space, which is generated by the elements of the Weil group. It states that if the elements of the basis algebra H• (n; C are described in terms of the Weil group it will result into a complicated description in terms of generators and relations may be simply unnecessary for computational purposes .

Let D be a division ring with centre F. Assume that M is a maximal subgroup of GLn(D) (n≥1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D:F]<∞. When D is non-commutative and F is infinite, it is also proved that if M satisfies a group identity and F[M] is algebraic over F, then we have either M=K* where K is a field and [D:F]<∞, or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GLn(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is abelian-by-finite. [ABSTRACT FROM AUTHOR]

The article presents a curious proof of Fermat's little theorem. Any formal power series can be represented in a unique way as a formal product where the coefficients being used are integers. A binomial theorem is not used in the proof of Fermat's little theorem. The author proves that any infinite sequence of integers has an inverse sequence of integers wherein all such sequences split into mutually inverse pairs. An inverse of a certain sequence for which the infinite product has a rich mathematical content is also expressed.