外FO-群

如果群 G 的所有真商群都是 FO-群,但是群 G 本身不是 FO-群,则称 G 为外 FO-群,本文将给出外 FO-群完全的结构描述．     

局部对称黎曼流形中的紧致极小子流形的Ricci曲率

设N ^n+p是截面曲率K_N 满足1/2 <δ≤ K_N≤ 1 的n+p维局部对称完备的δ-Pinching黎曼流形. Mⁿ是N^n+p 的紧致极小子流形. 该文讨论了这类子流形关于Ricci曲率有关的Pinching定理.     

Minimal 2-matching-covered graphs

A perfect 2-matching M of a graph G is a spanning subgraph of G such that each component of M is either an edge or a cycle. A graph G is said to be 2-matching-covered if every edge of G lies in some perfect 2-matching of G. A 2-matching-covered graph is equivalent to a “regularizable” graph, which was introduced and studied by Berge. A Tutte-type characterization for 2-matching-covered graph was given by Berge. A 2-matching-covered graph is minimal if G−e is not 2-matching-covered for all edges e of G. We use Berge’s theorem to prove that the minimum degree of a minimal 2-matching-covered graph other than K₂ and K₄ is 2 and to prove that a minimal 2-matching-covered graph other than K₄ cannot contain a complete subgraph with at least 4 vertices.     

On Long Minimal Zero Sequences in Finite Abelian Groups

Periodica Mathematica Hungarica -     

Minimal webs in Riemannian manifolds

For a given combinatorial graph G a geometrization (G, g) of the graph is obtained by considering each edge of the graph as a 1-dimensional manifold with an associated metric g. In this paper we are concerned with minimal isometric immersions of geometrized graphs (G, g) into Riemannian manifolds (N ⁿ , h). Such immersions we call minimal webs. They admit a natural ‘geometric’ extension of the intrinsic combinatorial discrete Laplacian. The geometric Laplacian on minimal webs enjoys standard properties such as the maximum principle and the divergence theorems, which are of instrumental importance for the applications. We apply these properties to show that minimal webs in ambient Riemannian spaces share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in such spaces. In particular we use appropriate versions of the divergence theorems together with the comparison techniques for distance functions in Riemannian geometry and obtain bounds for the first Dirichlet eigenvalues, the exit times and the capacities as well as isoperimetric type inequalities for so-called extrinsic R-webs of minimal webs in ambient Riemannian manifolds with bounded curvature.      

Centralizers of Involutions in Locally Finite Groups

The present article deals with locally finite groups G having an involution φ such that C _G (φ) is an SF-group. It is shown that G possesses a normal subgroup B which is a central product of finitely many groups isomorphic to PSL(2, K _i ) or SL(2, K _i ) for some infinite locally finite fields K _i of odd characteristic, such that [G, φ]′/B and G/[G, φ] are both SF-groups.     

Minimal projective resolutions

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the -algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the ``no loop' conjecture.

     

A Classification of Minimal Non-MNP-Groups

A finite group G is called an MNP-group if all maximal subgroups of every Sylow subgroup of G are normal in G. In this article, we give a complete classification of those groups which are not MNP-groups but all of whose proper subgroups are MNP-groups.     

Description of Barely Transitive Groups with Soluble Point Stabilizer

We consider a ring whose left modules of finite length are semisimple and prove some results on such a ring. We also consider when such a ring is a left V-ring.     

On Orderable Poly Engel Groups

It is known that an orderable n-Engel group is nilpotent. We show that an orderable group that is an extension of an n-Engel group by an n-Engel group is nilpotent-by-nilpotent. However, a finitely generated orderable poly n-Engel group need not be solvable in general.     

Groups whose cyclic subgroups have bounded permutability properties

It follows from classical results of Neumann and Macdonald that a group G has finite commuator subgroup if and only if either the normalizers of cyclic subgroups of G have boundedly finite indices or cyclic subgroups of G have bounded indices in their normal closures. In this paper, groups with a similar condition are considered, when normality is replaced by permutability.      

On locally finite -groups satisfying an Engel condition

For a given positive integer and a given prime number , let be the integer satisfying . We show that every locally finite -group, satisfying the -Engel identity, is (nilpotent of -bounded class)-by-(finite exponent) where the best upper bound for the exponent is either or if is odd. When the best upper bound is or . In the second part of the paper we focus our attention on -Engel groups. With the aid of the results of the first part we show that every -Engel -group is soluble and the derived length is bounded by some constant.

     

Infinite Groups with an Anticentral Element

An element of a group is called anticentral if the conjugacy class of that element is equal to the coset of the commutator subgroup containing that element. A group is called Camina group if every element outside the commutator subgroup is anticentral. In this paper, we investigate the structure of locally finite groups with an anticentral element. Moreover, we construct some non-periodic examples of Camina groups, which are not locally solvable.     

On a Commuting Graph on Conjugacy Classes of Groups

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.     

The Fitting Subgroup and Some Injectors of Radical Locally Finite Groups with min-p for All p

Abstract

This work was intended as an attempt to continue the study of the class ? of generalised nilpotent groups started in a previous paper. We present some results concerning the Fitting subgroup and the ?-injectors of a radical locally finite group satisfying min-p for all p.     

Minimal and locally minimal games and game forms

By Shapley’s (1964) theorem, a matrix game has a saddle point whenever each of its 2×2 subgames has one. In other words, all minimal saddle point free (SP-free) matrices are of size 2×2. We strengthen this result and show that all locally minimal SP-free matrices also are of size 2×2. In other words, if A is a SP-free matrix in which a saddle point appears after deleting an arbitrary row or column then A is of size 2×2. Furthermore, we generalize this result and characterize the locally minimal Nash equilibrium free (NE-free) bimatrix games.Let us recall that a two-person game form is Nash-solvable if and only if it is tight [V. Gurvich, Solution of positional games in pure strategies, USSR Comput. Math. and Math. Phys. 15 (2) (1975) 74-87]. We show that all (locally) minimal non-tight game forms are of size 2×2. In contrast, it seems difficult to characterize the locally minimal tight game forms (while all minimal ones are just trivial); we only obtain some necessary and some sufficient conditions. We also recall an example from cooperative game theory: a maximal stable effectivity function that is not self-dual and not convex.     

Minimal triangulations of graphs: A survey

Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.     

Perfect Category-Graded Algebras

In a perfect category, every object has a minimal projective resolution. We give a criterion for the category of modules over a category-graded algebra to be perfect.     

Constructing Minimal Projective Resolutions

Let Λ be a monomial algebra. For any Λ-module M, we describe graphically its minimal projective resolution as a weighted graph Δ(M). If M is finitely generated we give two algorithms for computing Δ(M). We also give a short computation-like argument to reprove a famous Syzygy Theorem.     

Minimal surfaces over stars

A JS surface is a minimal graph over a polygonal domain that becomes infinite in magnitude at the domain boundary. Jenkins and Serrin characterized the existence of these minimal graphs in terms of the signs of the boundary values and the side-lengths of the polygon. For a convex polygon, there can be essentially only one JS surface, but a non-convex domain may admit several distinct JS surfaces. We consider two families of JS surfaces corresponding to different boundary values, namely JS₀ and JS₁, over domains in the form of regular stars. We give parameterizations for these surfaces as lifts of harmonic maps, and observe that all previously constructed JS surfaces have been of type JS₀. We give an example of a JS₁ surface that is a new complete embedded minimal surface generalizing Scherk's doubly periodic surface, and show also that the JS₀ surface over a regular convex 2n-gon is the limit of JS₁ surfaces over non-convex stars. Finally we consider the construction of other JS surfaces over stars that belong neither to JS₀ nor to JS₁.