Verifying non-isomorphism of groups

2021 ◽  
Vol 105 (564) ◽  
pp. 467-473
Author(s):  
Des MacHale
Keyword(s):  
Group Theory ◽  
Commutative Diagram ◽  
Abstract Algebra ◽  
Image Position

The concept of isomorphism is central to group theory, indeed to all of abstract algebra. Two groups {G, *} and {H, ο}are said to be isomorphic to each other if there exists a set bijection α from G onto H, such that $$\left( {a\;*\;b} \right)\alpha = \left( a \right)\alpha \; \circ \;(b)\alpha $$ for all a, b ∈ G. This can be illustrated by what is usually known as a commutative diagram:

Minimal Regular Graphs of Girths Eight and Twelve

1966 ◽  
Vol 18 ◽  
pp. 1091-1094 ◽  
Author(s):  
Clark T. Benson
Keyword(s):  
Group Theory ◽  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

Homotopy Equivalence of a Cofibre Fibre Composite

1977 ◽  
Vol 29 (6) ◽  
pp. 1152-1156 ◽  
Author(s):  
Philip R. Heath
Keyword(s):  
Commutative Diagram ◽  
Fibre Composite ◽  
Topological Spaces ◽  
Homotopy Equivalence ◽  
Image Position

Consider the following commutative diagram in Top, the category of topological spacesin which j and j' are cofibrations, p and p' are (Hurewicz) fibrations and ƒ0, ƒi and ƒ2 are homotopy equivalences.

Lifting Problems and the Cohomology of C*-Algebras

1977 ◽  
Vol 29 (5) ◽  
pp. 1092-1111 ◽  
Author(s):  
Man-Duen Choi ◽  
Edward G. Effros
Keyword(s):  
Commutative Diagram ◽  
Linear Contraction ◽  
Linear Map ◽  
C Algebras ◽  
Image Position

Suppose that A and B are C*-algebras, J is a closed two-sided ideal in B, and that η: B →B/J is the quotient map. Given a linear contraction φ : A →B/J, a linear map Ψ: A →B is a lifting of φ if one has a commutative diagram

The Map SJ → SF Does Not Deloop Mod 2

1975 ◽  
Vol 27 (4) ◽  
pp. 737-745 ◽  
Author(s):  
Robert R. Clough
Keyword(s):  
Commutative Diagram ◽  
Image Position ◽  
Adams Conjecture

It has been widely conjectured that there exists a homotopy commutative Diagramwhere J is the stable Whitehead f-homomorphism and BSJ is the space constructed in [3]. In [4], Stasheff and the author proved that this conjecture is false. However, Quillen's proof of the Adams conjecture in [7] has as a corollary the existence of the homotopy commutative diagram

The Research and Design of Trusted Cloud Computing Platform Based on Group Theory

2013 ◽  
Vol 756-759 ◽  
pp. 867-871
Author(s):  
Ying Yang ◽  
Xue Hang Shao
Keyword(s):  
Cloud Computing ◽  
Group Theory ◽  
General Structure ◽  
Abstract Algebra ◽  
Public Key ◽  
Public Key Cryptosystems ◽  
Computing Platform ◽  
Cloud Computing Platform ◽  
The Cost ◽  
Research And Design

The cloud computing can greatly reduce the cost of computing, but is unable to ensure either the integrality or the confidentiality of data and calculation. Therefore, this paper considers of the safety of cloud computing, combining the thought of abstract algebra group theory in modern algebra, puts forward TCCPoGT (trusted cloud computing platform based on group theory) that designs in many different respects,such as general structure, public key cryptosystems and node management,etc. The analysis results show that the platform TCCPoGT can insure service security.

scholarly journals An Overview of the Foundations of the Hypergroup Theory

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1014
Author(s):  
Christos Massouros ◽  
Gerasimos Massouros
Keyword(s):  
Group Theory ◽  
Abstract Algebra ◽  
Special Issue ◽  
Abstract Group ◽  
Structural Relation ◽  
Fundamental Properties ◽  
Theory Versus ◽  
In The Beginning

This paper is written in the framework of the Special Issue of Mathematics entitled “Hypercompositional Algebra and Applications”, and focuses on the presentation of the essential principles of the hypergroup, which is the prominent structure of hypercompositional algebra. In the beginning, it reveals the structural relation between two fundamental entities of abstract algebra, the group and the hypergroup. Next, it presents the several types of hypergroups, which derive from the enrichment of the hypergroup with additional axioms besides the ones it was initially equipped with, along with their fundamental properties. Furthermore, it analyzes and studies the various subhypergroups that can be defined in hypergroups in combination with their ability to decompose the hypergroups into cosets. The exploration of this far-reaching concept highlights the particularity of the hypergroup theory versus the abstract group theory, and demonstrates the different techniques and special tools that must be developed in order to achieve results on hypercompositional algebra.

A campanological problem in group theory. II

1966 ◽  
Vol 62 (1) ◽  
pp. 11-18 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Group Theory ◽  
The Family ◽  
Image Position ◽  
Null Set

1. Let E be a finite non-null set and write (E) for the family of all permutations of E. Let be a non-null subset of (E) and write () for the subgroup of (E) generated by the members of . For any α ∈ we putso that () is a subgroup of () and is independent of the choice of α in . We suppose that E splits into k disjoint transitivity sets (orbits) Ei(1 ≤ i ≤ k) with respect to (); thus σEi = Ei for all σ ∈ ().

Trace in additive categories

1970 ◽  
Vol 67 (3) ◽  
pp. 541-547 ◽  
Author(s):  
Alan Thomas
Keyword(s):  
Abelian Group ◽  
Commutative Diagram ◽  
Additive Category ◽  
Definition Of ◽  
Image Position

1. Let ℳ be an additive category. (We refer to ((1), Ch. IX) for the definition of an additive category and associated terms. In particular a sequenceis exact if i = kerp and p = coker i.) We write End M for Hom(M, M). A trace on ℳ with values in an abelian group G is a collection of (abelian group) homomorphismsone for each M ∈ ℳ, satisfying the following two conditions:(i) Exactness. Given a commutative diagram with exact rows,then tA(f) + tC(h) = tB(g).

COMPONENTS AND MINIMAL NORMAL SUBGROUPS OF FINITE AND PSEUDOFINITE GROUPS

2019 ◽  
Vol 84 (1) ◽  
pp. 290-300
Author(s):  
JOHN S. WILSON
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Minimal Normal Subgroup ◽  
Normal Subgroups ◽  
First Order ◽  
Order Language ◽  
Image Position ◽  
A Finite Group

AbstractIt is proved that there is a formula$\pi \left( {h,x} \right)$in the first-order language of group theory such that each component and each non-abelian minimal normal subgroup of a finite groupGis definable by$\pi \left( {h,x} \right)$for a suitable elementhofG; in other words, each such subgroup has the form$\left\{ {x|x\pi \left( {h,x} \right)} \right\}$for someh. A number of consequences for infinite models of the theory of finite groups are described.

scholarly journals A Note on Absolutely Pure Modules

1976 ◽  
Vol 19 (3) ◽  
pp. 361-362 ◽  
Author(s):  
Edgar Enochs
Keyword(s):  
Commutative Diagram ◽  
Image Position ◽  
Finitely Presented ◽  
Exact Sequences

Fieldhouse observed that any finitely presented left R-module P is projective with respect to pure exact sequences, i.e.can always be completed to a commutative diagram when the sequence is pure exact. A left R-module A is absolutely pure if it is a pure submodule of every module which contains it.

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