scholarly journals Biorder-preserving coextensions of fundamental semigroups

1988 ◽  
Vol 31 (3) ◽  
pp. 463-467 ◽  
Author(s):  
David Easdown
Keyword(s):  
Group Theory ◽  
Semigroup Theory ◽  
Building Blocks ◽  
Inverse Semigroups ◽  
Extension Theory ◽  
Precise Definition ◽  
Simple Groups ◽  
Seminal Work ◽  
Fundamental Semigroups

In any extension theory for semigroups one must determine the basic building blocks and then discover how they fit together to create more complicated semigroups. For example, in group theory the basic building blocks are simple groups. In semigroup theory however there are several natural choices. One that has received considerable attention, particularly since the seminal work on inverse semigroups by Munn ([14, 15]), is the notion of a fundamental semigroup. A semigroup is called fundamental if it cannot be [shrunk] homomorphically without collapsing some of its idempotents (see below for a precise definition).

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

2010 ◽  
Vol 20 (07) ◽  
pp. 847-873 ◽  
Author(s):  
Z. AKHLAGHI ◽  
B. KHOSRAVI ◽  
M. KHATAMI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Prime Graphs ◽  
A Finite Group ◽  
Nonabelian Composition Factor

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

On the number of Sylow subgroups in finite simple groups

2020 ◽  
pp. 2150115
Author(s):  
Zhenfeng Wu
Keyword(s):  
Group Theory ◽  
Simple Group ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Sylow Subgroups

Denote by [Formula: see text] the number of Sylow [Formula: see text]-subgroups of [Formula: see text]. For every subgroup [Formula: see text] of [Formula: see text], it is easy to see that [Formula: see text], but [Formula: see text] does not divide [Formula: see text] in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712], we say that a group [Formula: see text] satisfies DivSyl(p) if [Formula: see text] divides [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text]. In this paper, we show that “almost for every” finite simple group [Formula: see text], there exists a prime [Formula: see text] such that [Formula: see text] does not satisfy DivSyl(p).

scholarly journals Congruences on free monoids and submonoids of polycyclic monoids

1993 ◽  
Vol 54 (2) ◽  
pp. 236-253 ◽  
Author(s):  
John Meakin ◽  
Mark Sapir
Keyword(s):  
Semigroup Theory ◽  
Free Monoid ◽  
Inverse Semigroups ◽  
Free Monoids ◽  
One To One

AbstractWe establish a one-to-one “group-like” correspondence between congruences on a free monoid X* and so-called positively self-conjugate inverse submonoids of the polycyclic monoid P(X). This enables us to translate many concepts in semigroup theory into the language of inverse semigroups.

COMBINATORIAL GROUP THEORY, INVERSE MONOIDS, AUTOMATA, AND GLOBAL SEMIGROUP THEORY

2002 ◽  
Vol 12 (01n02) ◽  
pp. 179-211 ◽  
Author(s):  
MANUEL DELGADO ◽  
STUART MARGOLIS ◽  
BENJAMIN STEINBERG
Keyword(s):  
Group Theory ◽  
Formal Language ◽  
Finite Semigroup ◽  
Semigroup Theory ◽  
Nilpotent Groups ◽  
Language Theory ◽  
Combinatorial Group Theory ◽  
Inverse Monoid ◽  
Group Presentation ◽  
Combinatorial Group

This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = <A|R> be a group presentation and ℬA, R its standard 2-complex. Suppose X is a 2-complex with a morphism to ℬA, R which restricts to an immersion on the 1-skeleton. Then we associate an inverse monoid to X which algebraically encodes topological properties of the morphism. Applications are given to separability properties of groups. We also associate an inverse monoid M(A, R) to the presentation <A|R> with the property that pointed subgraphs of covers of ℬA, R are classified by closed inverse submonoids of M(A, R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to be quasiconvex allowing semigroup theoretic variants on the usual proofs that the intersection of such subgroups is quasiconvex and that such subgroups are finitely generated. Generalizations are given to non-geodesic combings. We also obtain a formal language theoretic equivalence to quasiconvexity which holds even for groups which are not hyperbolic. Finally, we illustrate some applications of separability properties of relatively free groups to finite semigroup theory. In particular, we can deduce the decidability of various semidirect and Mal/cev products of pseudovarieties of monoids with equational pseudovarieties of nilpotent groups and with the pseudovariety of metabelian groups.

Genome Reconfiguration of Metamorphic Manipulators Based on Lie Group Theory

2008 ◽  
Author(s):  
Liping Zhang ◽  
Jian S. Dai
Keyword(s):  
Group Theory ◽  
Genetic Structure ◽  
Lie Group ◽  
Genetic Information ◽  
Evolutionary Process ◽  
Building Blocks ◽  
Biological Study ◽  
Lie Group Theory ◽  
Full Complement ◽  
Metamorphic Mechanisms

This paper investigates reconfiguration which was induced by topology change as a typical character of metamorphic mechanisms in a way analogous to the concept of genome varation in biological study. Genome is the full complement of genetic information that an organism inherits from its parents, espercially the set of genes they carry. Genome variation is to study the change and variation of this complement with genetic information and genes connectivity and is analogous to mechanisms reconfiguration of metamorphic mechanisms. Metamorphic mechanisms with reconfigurable topology are usually changing their configurations and varying mobility in accordance with different sub-working phase functions. The built-in spatial biological modules are for the first time compiled and introduced in this paper based on metamorphic building blocks in the form of metamorphic cells and associated inside break-down parts as the metamorphic genes for metamorphic bio-modeling as genome. The gene sequencing labels the genetic structure composition principle of the metamorphic manipulators. The bio-inspired mechanism configuration evolution is further introduced in this paper motivated by biological concept to metamorphic characteristics as different sub-phase working mechanisms gradually change and develop into different forms in a particular situation and over a period of time, as an evolutionary process of topological change that takes place over several motion phases during which a taxonomic group of organisms showing the change of their physical characteristics. Moreover, the proposed genetic structure composition principle in metamorphic manipulators leads to the development of module evolution and genetic operations based on the displacement subgroup algebraic properties of the Lie group theory. The topology transformations can further be simulated for configuration evolution and depicted with the genetic growth and degeneration in the living nature. Genome sequential reconfiguration for metamorphic manipulators promises to be mapped from degenerating the source generator to multiple sub-phase configurations. Evolution design illustrations are given to demonstrate the concept and principles.

scholarly journals Homogeneity of inverse semigroups

2018 ◽  
Vol 28 (05) ◽  
pp. 837-875 ◽  
Author(s):  
Thomas Quinn-Gregson
Keyword(s):  
Inverse Semigroup ◽  
Finite Groups ◽  
Abelian Groups ◽  
Semigroup Theory ◽  
Inverse Semigroups ◽  
Finitely Generated ◽  
Homogeneous Groups ◽  
Inverse Subsemigroups

An inverse semigroup [Formula: see text] is a semigroup in which every element has a unique inverse in the sense of semigroup theory, that is, if [Formula: see text] then there exists a unique [Formula: see text] such that [Formula: see text] and [Formula: see text]. We say that a countable inverse semigroup [Formula: see text] is a homogeneous (inverse) semigroup if any isomorphism between finitely generated (inverse) subsemigroups of [Formula: see text] extends to an automorphism of [Formula: see text]. In this paper, we consider both these concepts of homogeneity for inverse semigroups, and show when they are equivalent. We also obtain certain classifications of homogeneous inverse semigroups, in particular periodic commutative inverse semigroups. Our results may be seen as extending both the classification of homogeneous semilattices and the classification of certain classes of homogeneous groups, such as homogeneous abelian groups and homogeneous finite groups.

scholarly journals Nice Efficient Presentions for all Small Simple Groups and their Covers

2004 ◽  
Vol 7 ◽  
pp. 266-283 ◽  
Author(s):  
Colin M. Campbell ◽  
George Havas ◽  
Colin Ramsay ◽  
Edmund F. Robertson
Keyword(s):  
Group Theory ◽  
Power Relations ◽  
Simple Groups ◽  
Computational Group Theory ◽  
The Status ◽  
Coset Enumeration ◽  
Covering Groups ◽  
Computational Properties ◽  
Selection Of ◽  
Better Than

AbstractPrior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors‘presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.

scholarly journals A ŠVARC–MILNOR LEMMA FOR MONOIDS ACTING BY ISOMETRIC EMBEDDINGS

2011 ◽  
Vol 21 (07) ◽  
pp. 1135-1147 ◽  
Author(s):  
ROBERT GRAY ◽  
MARK KAMBITES
Keyword(s):  
Group Theory ◽  
Cayley Graph ◽  
Semigroup Theory ◽  
Geometric Group Theory ◽  
Distance Functions ◽  
Cancellative Monoid ◽  
Isometric Embeddings ◽  
Key Techniques

We continue our program of extending key techniques from geometric group theory to semigroup theory, by studying monoids acting by isometric embeddings on spaces equipped with asymmetric, partially defined distance functions. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Švarc–Milnor lemma to this setting.

A new definition of conjugacy for semigroups

2018 ◽  
Vol 17 (02) ◽  
pp. 1850032 ◽  
Author(s):  
Janusz Konieczny
Keyword(s):  
Group Theory ◽  
Inverse Semigroup ◽  
Directed Graphs ◽  
Conjugacy Classes ◽  
Inverse Semigroups ◽  
Transformation Semigroups ◽  
Arbitrary Semigroup ◽  
Basic Transformation ◽  
Definition Of ◽  
Natural Way

The conjugacy relation plays an important role in group theory. If [Formula: see text] and [Formula: see text] are elements of a group [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] for some [Formula: see text]. The group conjugacy extends to inverse semigroups in a natural way: for [Formula: see text] and [Formula: see text] in an inverse semigroup [Formula: see text], [Formula: see text] is conjugate to [Formula: see text] if [Formula: see text] and [Formula: see text] for some [Formula: see text]. In this paper, we define a conjugacy for an arbitrary semigroup [Formula: see text] that reduces to the inverse semigroup conjugacy if [Formula: see text] is an inverse semigroup. (None of the existing notions of conjugacy for semigroups has this property.) We compare our new notion of conjugacy with existing definitions, characterize the conjugacy in basic transformation semigroups and their ideals using the representation of transformations as directed graphs, and determine the number of conjugacy classes in these semigroups.

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