Isomorphic factorizations of complete equipartite graphs

A complete equipartite graph is a complete multipartite graph such that the color classes are equinumerous. An isomorphic factorization of a graph is a partition of the line set into disjoint isomorphic parts. It is shown that an isomorphic factorization of the complete equipartite graph into t isomorphic subgraphs exists whenever the total number of lines is divisible by t.     

On Modules Which are Isomorphic to Relatively Divisible or Pure Submodules of Each Other

Abstract

In this note, we provide a generalization of a well-known result of module theory which states that two injective modules are isomorphic when they are isomorphic to submodules of each other. More precisely, we show here that two RD-injective (respectively, pure-injective) modules over an integral domain are isomorphic if they are isomorphic to relatively divisible (respectively, pure) sub- modules of each other.     

On Cantor-Bernstein type theorems in Riesz spaces

We generalize the main result of [21] to Riesz spaces. Let X and Y be Riesz spaces with σ-complete Boolean algebras of projection bands. If X and Y are each Riesz isomorphic to a projection band of the other space then the spaces are Riesz isomorphic. As an application of the above theorem we give an example of non-Riesz isomorphic Banach lattices such that: (1) their order (= topological) duals are Riesz isomorphic and (2) each of them is Riesz isomorphic to a projection band of the other one.     

Maximal nests in the Calkin algebra

We prove that if two countable commutative lattices of projections in the Calkin algebra are order isomorphic, then they are unitarily equivalent. We show that there are isomorphic maximal nests of projections in the Calkin algebra that are order isomorphic but not similar. Assuming the continuum hypothesis, we show that all maximal nests of projections in the Calkin algebra are order isomorphic.

     

The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.     

On solutions to the Schroeder-Bernstein problem for Banach spaces

We investigate the geometry of the Banach spaces failing Schroeder-Bernstein Property (SBP). Initially we prove that every complex hereditarily indecomposable Banach space H is isomorphic to a complemented subspace of a Banach space S(H) that fails SBP in such a way that the only complemented hereditarily indecomposable subspaces of S(H) are those which are nearly isomorphic to H. Then we show that every Banach space having Mazur property is isomorphic to some complemented subspace of a Banach space which is not isomorphic to its square but isomorphic to its cube. Finally, we prove that if a Banach space X fails SBP then either it is not primary or the Grothendieck group K₀(L(X)) of the algebra of operators on X is not trivial.     

To what extent does the dual Banach spaceE′ determine the polynomials overE?

We show that under conditions of regularity, ifE′ is isomorphic toF′, then the spaces of homogeneous polynomials overE andF are isomorphic. Some subspaces of polynomials more closely related to the structure of dual spaces (weakly continuous, integral, extendible) are shown to be isomorphic in full generality.     

关于Grothendieck群的几点注记

作者在本文中围绕Ｇｒｏｔｈｅｎｄｉｅｃｋ群对几个问题进行了讨论，主要结果有：１．一切有限生成Ｒ－模的同构类作成一个集合；２．在任意由Ｒ－模作成的集合中稳定同的关系是合同关系；３．Ｇｒｏｔｈｅｎｄｉｅｃｋ群的同构不变性成立。     

A family of nonisomorphic Markov random fields

It was recently shown that there exists a family of ℤ² Markov random fields which areK but are not isomorphic to Bernoulli shifts [4]. In this paper we show that most distinct members of this family are not isomorphic. This implies that there is a two parameter family of ℤ² Markov random fields of the same entropy, no two of which are isomorphic.     

Uniqueness of preduals of certain Banach spaces

In [1], the authors have shown the existence of non-quasireflexive Banach spaces having unique isomorphic preduals. In fact, certain James-Lindenstrauss’ spaces have this property. In this paper it is shown that there are many such separable spaces. More precisely, there exist infinitely many different isomorphic types of James-Lindenstrauss’ spaces which are non-quasireflexive and have unique isomorphic preduals. The research of both authors was partially supported by N.S.F. Grant No. MPS75-07115     

Strongly self-complementary and hereditarily isomorphic tournaments

A tournament is strongly self-complementary if every subtournament is self-complementary. TournamentsR, B defined on the same vertex setV are hereditarily isomorphic if for every subsetA ofV, the induced subtournamentsR(A) andB(A) are isomorphic. We determine all strongly self-complementary tournaments and apply this result to characterize the paris of hereditarily isomorphic tournaments.     

Strongly self-complementary and hereditarily isomorphic tournaments

A tournament is strongly self-complementary if every subtournament is self-complementary. TournamentsR, B defined on the same vertex setV are hereditarily isomorphic if for every subsetA ofV, the induced subtournamentsR(A) andB(A) are isomorphic. We determine all strongly self-complementary tournaments and apply this result to characterize the paris of hereditarily isomorphic tournaments.     

Automorphism group and dimension of ordered sets

It is shown that a finite groupG is isomorphic to the automorphism group of a two-dimensional ordered set if and only if it is a generalized wreath product of symmetric groups over an ordered index set that is a dual tree. Furthermore, every finite abelian group is isomorphic to the full automorphism group of a three-dimensional ordered set. Also every finite group is isomorphic to the automorphism group of an ordered set that does not contain an induced crown with more than four elements.     

Two maximal subgroups of E8(2)

We show that E₈(2) has a unique conjugacy class of subgroups isomorphic to PSp₄(5) and a unique conjugacy class of subgroups isomorphic to PSL₃(5). There normalizers are maximal subgroups of E₈(2) and are, respectively, isomorphic to PGSp₄(5) and Aut(PSL₃(5)).     

Tensor roots of isomorphisms and weak limits of transformations

We prove that two transformations S and T are isomorphic if their Cartesian squares are isomorphic, under the assumption that the sequence of powers of T converges weakly to a polynomial.     

Banach spaces whose duals are isomorphic to l1(Γ)

Banach spaces X whose duals are isomorphic or isometric to l₁(Γ) are characterized by certain classes of operators on X. It is proved that a separable, conjugate space isomorphic to a complemented subspace of an L₁(S, Σ, μ) space is isomorphic to l₁; a $\text{L}$₁ space contained in a separable, conjugate space is isomorphic to a subspace of l₁.     

On the structure of compact torsion groups

It is shown that every profinite torsion group has a finite series of closed characteristic subgroups in which each factor either is a pro-p-group for some primep or is isomorphic to a Cartesian product of isomorphic finite simple groups.     

Conditions ensuring starter induced 1‐factorizations are not isomorphic

In general, it is difficult to determine whether two starter induced 1‐factorizations of K_2n are isomorphic. However, when one of the 1‐factorizations has a unique starter group to within conjugacy, we show that two starter induced 1‐factorizations on K_2n are isomorphic, if and only if, the corresponding starters are isomorphic. Two starters are isomorphic if there is a group isomorphism between the respective starter groups which takes one starter to the other. It is relatively easy to check whether starters are isomorphic in many examples. The difficulty comes in showing there is a unique starter group to within conjugacy. A number of sufficient conditions for this are given; one such condition is the assumption that the 1‐factorization be irreducible, a condition which applies to all almost perfect 1‐factorizations. Also, generalized Mullin Nemeth starters are introduced and discussed in this context. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 124–143, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10023     

Substandard models of finite set theory

A survey of the isomorphic submodels of V_ω, the set of hereditarily finite sets. In the usual language of set theory, V_ω has 2^?0 isomorphic submodels. But other set‐theoretic languages give different systems of submodels. For example, the language of adjunction allows only countably many isomorphic submodels of V_ω (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)