Solvable D_2-Groups

Let G be a finite group, Irr_1(G) be the set of nonlinear irreducible characters of G and cd_1(G) the set of degrees of the characters in Irr_1(G). A group G is said to be a D_2-group if |cd_1(G)| =|Irr_1(G)|-2. In this paper, we give a complete classification of solvable D_2-groups.     

Surfaces which contain helical geodesics

LetM be a complete smooth surface of constant mean curvature in the Euclidean 3-space. If there exists two helical geodesics onM through each point ofM, thenM is either a plane, a sphere, or a circular cylinder.     

Transitive permutation groups in which all derangements are involutions

Let G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.     

Strongly Real Special 2-Groups

Strongly real groups and totally orthogonal groups form two important subclasses of real groups. In this article we give a characterization of strongly real special 2-groups. This characterization is in terms of quadratic maps over fields of characteristic 2. We then provide examples of groups which are in one subclass and not the other. It is a conjecture of Tiep that such examples are not possible for finite simple groups.     

n2-sets in a projective plane which determine exactly n2+n lines

In this paper n²-sets, in a projective plane, determining exactly n²+n lines are completely characterized.Research partially supported by G.N.S.A.G.A. (C.N.R.).Paper written while the second author was at the University of Rome supported by a grant within the Italian Hungarian cultural exchange program.     

Measurable Categories and 2-Groups

Using the theory of measurable categories developed in [10], we provide a notion of representations of 2-groups better suited to physically and geometrically interesting examples than that using 2-VECT (cf. [8]). Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in [6], and suggest state-integral constructions for 4-manifold invariants.     

Representations of 2-Groups by Polynomials

It is known that any finite p-group can be represented by polynomials. However, how to represent p-groups and how to classify p-groups up to isomorphism are interesting and open questions. In this article, we investigate the 2-groups of order 8, and represent the dihedral group D_2n, the generalized quaternion group Q_2n, and the infinite dihedral group D.2000 Mathematics Subject Classification: 20C99, 20E99     

2-Groups with normal noncyclic subgroups

We study nearly-hamiltonian 2-groups (¯H₂-groups). A nonabelian 2-group having at least one proper noncyclic subgroup is called an ¯H₂-group if all noncyclic subgroups are normal.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 75–84, July, 1968.The author expresses his deep gratitude towards Professor S. N. Chernikov for guidance while writing this paper.     

The Neighbourhood of Dihedral 2-Groups

We examine two particular constructions that derive from a 2-group G=G() another 2-group G(*) for the case when G() is one of , , . The constructions (cyclic and dihedral) have the property that x*y=xy for exactly 3/4 of all pairs (x,y)G×G. Work supported by by Grant Agency of Charles University, grant number 269/2001/B-MAT/MFF. The second and the third authors were also supported by the institutional grants MSM 113200007 and MSM 210000010, respectively.Mathematics Subject Classifications (2000) primary: 20D60, secondary: 05B15.     

Semigroups which contain magnifying elements are factorizable

A semigroup S is factorizable if it contains two proper subsemigroups A and B such that S = AB. An element a of a semigroup 5 is a left ( resp. right) magnifier if there exists a proper subset M of S such that S = aM (resp. S - Ma).

In this paper we prove that every semigroup containing magnifying elements is factorizable. Thus we solve a problem raised up by F. Catino and F. Migliorini in [2], namely to find necessary and sufficient conditions in order that a semigroup with magnifying elements be factorizable. Partial answers to this problem have been obtained by K. Tolo ([14]), F. Catino and F. Migliorini ([2]), for semigroups with left magnifiers and which are regular or have left units or right magnifiers, by V. M. Klimov ([9]), for Baer-Levi and Croisot-Teissier semigroups, and by M. Gutan ([4]), for right cancellative, right simple, idempotent free semigroups.     

Finite geometries which contain dual affine planes

Finite geometries in which each plane is projective or dual affine over the field of two elements, or affine over the field of three elements, are studied. It is shown that no connected geometry can mix all three species of planes, and the geometries in which projective and dual affine planes occur are classified.