236 research outputs found
Properties of the Generalized Zig-Zag Product of Graphs
The operation of zig-zag products of graphs is the analogue of the semidirect
product of groups. Using this observation, we present a categorical description
of zig-zag products in order to generalize the construction for the category of
simple graphs. Also, we examine the covering properties of zig-zag products and
we utilize these results to estimate their spectral invariants in general. In
addition, we provide specific spectral analysis for some such products.Comment: 15 page
Factor groups, semidirect product and quantum chemistry
In this paper we prove some general theorems about representations of finite
groups arising from the inner semidirect product of groups. We show how these
results can be used for standard applications of group theory in quantum
chemistry through the orthogonality relations for the characters of irreducible
representations. In this context, conditions for transitions between energy
levels, projection operators and basis functions were determined. This approach
applies to composite systems and it is illustrated by the dihedral group
related to glycolate oxidase enzyme
On a Generalization of the Notion of Semidirect Product of Groups
We introduce an external version of the internal r-fold semidirect product of
groups (SDP) of Carrasco and Cegarra. Just as for the classical external SDP,
certain algebraic data are required to guarantee associativity of the
construction. We give an algorithmic procedure for computing axioms
characterizing these data. Additionally, we give criteria for determining when
a family of homomorphisms from the factors of an SDP into a monoid or group
assemble into a homomorphism on the entire SDP. These tools will be used
elsewhere to give explicit algebraic axioms for hypercrossed complexes, which
are algebraic models for classical homotopy types introduced by Carrasco and
Cegarra
The multiple holomorph of a semidirect product of groups having coprime exponents
Given any group , the multiple holomorph is the
normalizer of the holomorph
in the group of all permutations of , where denotes the right regular
representation. The quotient has order
a power of in many of the known cases, but there are exceptions. We shall
give a new method of constructing elements (of odd order) in when
, where is a group of finite exponent coprime to and
is the cyclic group of order .Comment: 12 page
Amenability and Inner Amenability of Transformation Groups
In this paper, we show that there is a net for amenable transformation groups
like F{\o}lner net for amenable groups and investigate amenability of a
transformation group constructed by semidirect product of groups. We introduce
inner amenability of transformation groups and characterize this property
It\^o's theorem and metabelian Leibniz algebras
We prove that the celebrated It\^{o}'s theorem for groups remains valid at
the level of Leibniz algebras: if is a Leibniz algebra such that
, for two abelian subalgebras and , then
is metabelian, i.e. . A structure type theorem for
metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras
having the derived algebra of dimension are described, classified and their
automorphisms groups are explicitly determined as subgroups of a semidirect
product of groups
associated to any vector space .Comment: Final version; to appear in Linear Multilinear Algebr
A remark on MAKE -- a Matrix Action Key Exchange
In a recent paper [arXiv:2009.00716], Rahman and Shpilrain proposed a new
key-exchange protocol MAKE based on external semidirect product of groups. The
purpose of this paper is to show that the key exchange protocol is insecure. We
were able to break their challenge problem in under a second
Twisted product of Lie groups
In this article we define the twisted product of groups as the generalization
of the semidirect product of groups. We will find the necessary and sufficient
condition in order that the twisted product of groups to be a group. In
particular, for two copies of the same group, the twisted product of group by
itself through the action of inner automorphisms is a group if and only if the
initial group is a metabelian group. Further we will construct Lie algebra for
Lie group of a twisted product of Lie groups. In the case of twisted product of
Lie group by itself by means of the action of inner automorphisms we find the
dependence of the scalar curvature for resulting Lie group on the scalar
curvature for initial Lie group.Comment: 20 pages, LATEX, to be published in Siberian Mathematical Journa
Entropy waves, the zig-zag graph product, and new constant-degree
The main contribution of this work is a new type of graph product, which we
call the {\it zig-zag product}. Taking a product of a large graph with a small
graph, the resulting graph inherits (roughly) its size from the large one, its
degree from the small one, and its expansion properties from both! Iteration
yields simple explicit constructions of constant-degree expanders of arbitrary
size, starting from one constant-size expander.
Crucial to our intuition (and simple analysis) of the properties of this
graph product is the view of expanders as functions which act as ``entropy
wave" propagators -- they transform probability distributions in which entropy
is concentrated in one area to distributions where that concentration is
dissipated. In these terms, the graph products affords the constructive
interference of two such waves.
Subsequent work [ALW01], [MW01] relates the zig-zag product of graphs to the
standard semidirect product of groups, leading to new results and constructions
on expanding Cayley graphs.Comment: 31 pages, published versio
Loops and Semidirect Products
A \emph{loop} is a set together with a binary operation
such that (i) for each , the left and right translation
mappings and are bijections, and (ii) there exists a two-sided identity element .
Thus loops can be thought of as "nonassociative groups". In this paper we study
standard, internal and external semidirect products of loops with groups. These
are generalizations of the familiar semidirect product of groups.Comment: 27 pages, LaTeX2e, uses tcilatex.sty; final version; to appear in
Comm. Algebr
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