Narrow Results

The conjugacy classes of any group are important since they reflect some aspects of the structure of the group. The construction of the conjugacy classes of finite groups has been a subject of research for several authors. Let n,m be positive integers and be the direct product of m copies of the symmetric group S_n of degree n. Then is a subgroup of the symmetric group S_mn of degree m × n. Let g∈S_mn, of type [mⁿ] where each m-cycle contains one symbol from each set of symbols in that order on which the copies of S_n act. Then g permutes the elements of the copies of S_n in and generates a cyclic group C_m = 〈g〉 of order m. The wreath product of S_n with C_m is a split extension or semi-direct product of by C_m, denoted by . It is clear that is a subgroup of the symmetric group S_mn. In this paper we give a method similar to coset analysis for constructing the conjugacy classes of , where m is prime. Apart from the fact that this is an alternative method for constructing the conjugacy classes of the group , this method is useful in the construction of Fischer–Clifford matrices of the group . These Fischer–Clifford matrices are useful in the construction of the character table of .

In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups $G$ are fully characterized in the class of all groups by the set $tp(G)$ of types realized in $G$. Furthermore, it turns out that these groups $G$ are fully characterized already by some particular rather restricted fragments of the types from $tp(G)$. In particular, every finitely generated nilpotent group is completely defined by its ${\exists }^{+}$-types, while a finitely generated rigid group is completely defined by its $\forall$-types, and a finitely generated metabelian or polycyclic group is completely defined by its $\forall \exists$-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number $F$, genus $g$ and type $t$. It is known that for any numerical semigroup $\frac{g}{F+1-g}\le t\le 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with $t=\frac{g}{F+1-g}$. We show that for a fixed $\alpha$ the number of numerical semigroups with Frobenius number $F$ and type $F-\alpha$ is eventually constant for large $F$. The number of numerical semigroups with genus $g$ and type $g-\alpha$ is also eventually constant for large $g$.

We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP [The GAP Group, GAP — Groups, Algorithms and Programming, Version 4.8.6; 2016, https://www.gap-system.org] package NumericalSgps [M. Delgado and P. A. García-Sánchez and J. Morais, “numericalsgps”: A GAP package on numerical semigroups, https://github.com/gap-packages/numericalsgps].

The aim of this paper is to give a new characterization of cyclic modules over local rings, namely, we show that if $N$ is a nonzero finitely generated module over a local (Noetherian) ring $(R,𝔪)$ such that for some integer $n\ge 1$, the submodule ${𝔪}^{n}N$ of $N$ is cyclic, then every submodule of $N$ is cyclic and the ring $R/{\mathrm{\text{Ann}}}_R(N)$ is a $\mathrm{\text{D}}VR$, whenever $\mathrm{\text{depth}}N>0$. As a consequence, it follows that $R/{\mathrm{\text{Ann}}}_R(N/{\mathrm{\Gamma }}_{𝔪}(N))$ is a $\mathrm{\text{D}}VR$ and every submodule of $N/{\mathrm{\Gamma }}_{𝔪}(N)$ is cyclic.

In this paper, we carry out a fairly comprehensive study of two special classes of numerical semigroups, one generated by the sequence of partial sums of an arithmetic progression and the other one generated by a shifted geometric progression, in embedding dimension $4$. Both these classes have the common feature that they have unique expansions of the Apéry set elements.

Given two numerical semigroups S and S', the distance between S and S' is the cardinality of S\S' plus the cardinality of S'\S. In this paper we study those numerical semigroups S for which there is a symmetric numerical semigroup whose distance to S is one.

In this chapter, E-W-trending tectonic systems are introduced, including those in the Arctic region, China, the Northern Hemisphere and the Southern Hemisphere.

In this chapter, N-S-trending tectonic systems are introduced, including those in the Ural Mountains in Asia, the island of Sakhalin in Russia, the Andes Mountains in South America, East Africa, the oceans and China.

In this chapter, N-E-trending tectonic systems are introduced, including those in China, New Zealand in the Pacific, northwestern Europe, the eastern United States, eastern South America and eastern Africa.

In this chapter, N-N-E-trending tectonic systems are introduced, including those in China, New Zealand–Tonga, the Eastern United States and the east coast of South America.

In this chapter, N-W-trending tectonic systems are introduced, including those in China, central Asia, the North Caucasus, the Zagros of the middle East, the western coast of North America and the Gulf of Suez.

In this chapter, epsilon-shaped tectonic systems are introduced, including those in China, Eurasia, Irkutsk in southern Siberia of Russia, Teli in Turkey, Gadomein France, England, North America, Cincinnati in southern North America and Brazil in South America.

In this chapter, S-shaped or reverse S-shaped tectonic systems are introduced, including reverse S-shaped systems in Qinghai–Tibet–Burma, China, the western coast of North America and S-shaped systems in western South America and western Africa.

In this chapter, rotation-torsional tectonic systems are introduced, including the rotation-torsional systems in China, the northern Sakhalin geese-shaped systems in northeastern Russia, the broom-shaped systems in the western Indian Ocean and southwestern Pacific, the double-ring compound rotation-torsional systems in Antarctica and the concentric radial systems in the Arctic.

Let L_S denote the language of (right) S-acts over a monoid S and let Σ_S be a set of sentences in L_S which axiomatises S-acts. A general result of model theory says that Σ_S has a model companion, denoted by T_S, precisely when the class of existentially closed S-acts is axiomatisable and in this case, T_S axiomatises . It is known that T_S exists if and only if S is right coherent. Moreover, by a result of Ivanov, T_S has the model-theoretic property of being stable.

In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over T_S algebraically, thus reducing our examination of T_S to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that T_S is stable and to prove another result of Ivanov, namely that T_S is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that T_S is totally transcendental and is such that the U-rank of any type coincides with its Morley rank.