Multiple Holomorphs of Dihedral and Quaternionic Groups

The holomorph of a group G is Norm _B (λ(G)), the normalizer of the left regular representation λ(G) in its group of permutations B = Perm(G). The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group λ(G) and its conjugates within the holomorph. We explore the multiple holomorphs of the dihedral groups D _n and quaternionic (dicyclic) groups Q _n for n ≥ 3.     

Completely Decomposable Quotient Divisible Abelian Groups with Isomorphic Endomorphism Semigroups

Mathematical Notes - Let $$\Lambda$$ be a class of Abelian groups. A group $$A\in\Lambda$$ is said to be determined by its endomorphism semigroup $$E^\star(A)$$ in the class $$\Lambda$$ if every...     

Almost Isomorphic Torsion-Free Abelian Groups and Similarity of Homogeneously Decomposable Groups

In the paper the concept of the similarity of homogeneously decomposable groups is introduced. One investigates the cases in which almost isomorphic groups are similar or isomorphic. Mathematics Subject Classification (2000) 20K99.     

The Normalizer Property for Integral Group Rings of Holomorphs of Finite Groups with Trivial Center

Let G=Hol(H) be the holomorph of a finite group H. If there is a prime q dividing |H| such that every q-central automorphism of H is inner and Z(H)=1, then every Coleman automorphism of G is inner. In particular, the normalizer property holds for G.     

A Note on Isomorphic Groups and Nonhomeomorphic Spaces

Given a Tychonov space X we can construct another space Y with the same group of homeomorphisms such that X and Y are, in some sense, "almost arbitrarily different".     

Isomorphic Commutative Group Algebras of p-Mixed Warfield Groups

Let G be a p-mixed Warfield Abelian group and F a field of char F = p ≠ 0. It is proved that if for any group H the group algebras FH and FG are F-isomorphic, then H is isomorphic to G. This presentation enlarges a result of W. May argued when G is p-local Warfield Abelian and published in Proc. Amer. Math. Soc. （1988）.     

On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFI-group is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13 Grinshpon , S. Ya. , Nikolskaya (Savinkova) , M. M. ( 2011 ). Fully invariant subgroups of abelian p-groups with finite Ulm-Kaplansky invariants . Commun. Algebra 39 : 4273 – 4282 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and [14 Grinshpon , S. Ya. , Nikolskaya (Savinkova) , M. M. ( 2011-2012/2014 ). Torsion IF-groups . Fundam. Prikl. Mat. 17 : 47 – 58 ; translated in J. Math. Sci. 197:614–622 . [Google Scholar]].     

Sets of Transvections Generating Subgroups Isomorphic to Special Linear Groups

ABSTRACT

The main result of this paper is a graph-theoretic necessary and sufficient condition, for a given set of transvections in SL(n, K) (n > 2 and K a finite field of characteristic not 2 or 3), to generate a group isomorphic to SL (m, L), for some m and some subfield L of K.     

Primary Abelian Groups Isomorphic to the Proper Fully Invariant Subgroup

We study primary Abelian groups containing proper fully invariant subgroup isomorphic to the group. The admissable sequence of the Ulm–Kaplansky invariants for primary groups is used. Using these sequences, IF-groups are described in some important classes of Abelian p-group.     

On Groups that are Isomorphic with Every Subgroup of Finite Index and their Topology

The main result is that a finitely generated group that is isomorphicto all of its finite index subgroups has free Abelian firsthomology, and that its commutator subgroup is a perfect group.A number of corollaries on the structure of such groups areobtained, including a method of constructing all such groupsfor which the commutator subgroup has a trivial centralizer.As an application, conditions are presented for the coveringspaces of compact manifolds that determine when the fundamentalgroups of the base spaces are free Abelian.     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group.E is a holomorphic vectorbundle of rank r with trivial pull-back to W=C~n-{0}.We prove the existence of a non-vanishingsection of L(?)E for some line bundle on X and study the vector bundles filtration structure of E.These generalize the results of D.Mall about structure theorem of such a vector bundle E.     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group. E is a holomorphic vector bundle of rank r with trivial pull-back to W = ℂ ⁿ –{0}. We prove the existence of a non-vanishing section of L⊗E for some line bundle on X and study the vector bundles filtration structure of E. These generalize the results of D. Mall about structure theorem of such a vector bundle E. The research was supported by 973 Project Foundation of China and the Outstanding Youth Science Grant of NSFC (grant no. 19825105)     

Isometries with Isomorphic Invariant Subspace Lattices

J. B. Conway and T. A. Gillespie (J. Funct. Anal.64 (1985), 178–189) characterized those reductive normal operators which have isomorphic invariant subspace lattices. In a subsequent paper (J. Operator Theory22 (1989), 31–49) they gave several necessary conditions of isomorphism in the class of nonreductive isometries. In this paper, we provide a new necessary condition when the isometry contains a bilateral shift. Furthermore, we give complete characterization if the nonreductive components of the isometries are cyclic. It turns out that this characterization is of different types in the unitary and in the nonunitary case. We describe also when absolutely continuous unitary operators have spatially isomorphic invariant subspace lattices. Our results provide answers for questions posed in the second Conway and Gillespie paper referenced above.     

The Coarea Formula for Vector Functions on Carnot Groups with Sub-Lorentzian Structure

Siberian Mathematical Journal - We establish the coarea formula as an expression for the measure of a subset of a Carnot group in terms of the sub-Lorentzian measure of...     

Invariant Minimal Unit Vector Fields on Lie Groups

We provide a new characterization of invariant minimal unit vector fields on Lie groups and use it to construct some new examples. In particular, we determine all these vector fields on three-dimensional Lie groups.     

On a Problem of P. Hall about Groups Isomorphic to all their Non-Trivial Normal Subgroups

A scheme of construction of infinite groups, other than simplegroups, free groups of infinite rank and the infinite cyclicgroup, which are isomorphic to all their non-trivial normalsubgroups is presented. Some results about the automorphismgroups of simple infinite groups are also obtained. In particular,it is proved that there is an infinite group G of any sufficientlylarge prime exponent p (or which is torsion-free) all of whoseproper subgroups are cyclic, and such that the groups Aut Gand Out G are isomorphic. The proofs use the technique of gradeddiagrams developed by A. Yu. Ol'shanskii. 1991 Mathematics SubjectClassification: 20F05, 20F06.     

Isomorphic extension theorem

A sufficient condition is obtained for the isomorphic-extension theorem in a category to hold. It is shown by an example that this is not a necessary condition.Translated from Matematicheskie Zametki, Vol. 7, No. 6, pp. 683–686, June, 1970.     

Isomorphic Steiner symmetrization

This paper proves that there exist 3n Steiner symmetrizations that transform any convex set Kⁿ into an isomorphic Euclidean ball; i.e. if vol(K)=vol(D_n) where D_n is the standard Euclidean unit ball, then K can be transformed into a body K such that c₁D_nKc₂D_n, where c₁,c₂ are numerical constants. Moreover, for any c>2, cn symmetrizations are also enough.     

Isomorphic scheduling problems

We consider general properties of isomorphic scheduling problems that constitute a new class of pairs of mutually related scheduling problems. Any such a pair is composed of a scheduling problem with fixed job processing times and its time-dependent counterpart with processing times that are proportional-linear functions of the job starting times. In order to introduce the class formally, first we formulate a generic scheduling problem with fixed job processing times and define isomorphic problems by a one-to-one transformation of instances of the generic problem into instances of time-dependent scheduling problems with proportional-linear job processing times. Next, we prove basic properties of isomorphic scheduling problems and show how to convert polynomial algorithms for scheduling problems with fixed job processing times into polynomial algorithms for proportional-linear counterparts of the original problems. Finally, we show how are related approximation algorithms for isomorphic problems. Applying the results, we establish new worst-case results for time-dependent parallel-machine scheduling problems and prove that many single- and dedicated-machine time-dependent scheduling problems with proportional-linear job processing times are polynomially solvable.