scholarly journals A DESCRIPTION OF A CLASS OF FINITE SEMIGROUPS THAT ARE NEAR TO BEING MAL'CEV NILPOTENT

2013 ◽  
Vol 12 (05) ◽  
pp. 1250221 ◽  
Author(s):  
E. JESPERS ◽  
M. H. SHAHZAMANIAN
Keyword(s):  
Finite Group ◽  
Algebraic Structure ◽  
Disjoint Union ◽  
Finite Semigroup ◽  
Local Information ◽  
Connected Components ◽  
Global Information ◽  
Nilpotent Ideal ◽  
Finite Semigroups ◽  
A Finite Group

In this paper we continue the investigation on the algebraic structure of a finite semigroup S that is determined by its associated upper non-nilpotent graph [Formula: see text]. The vertices of this graph are the elements of S and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Mal'cev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of pseudo-nilpotent semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. some two-generated subsemigroups are not nilpotent. It turns out that a finite monoid (in particular, a finite group) is pseudo-nilpotent if and only if it is nilpotent. Our main result is a description of pseudo-nilpotent finite semigroups S in terms of their associated graph [Formula: see text]. In particular, S has a largest nilpotent ideal, say K, and S/K is a 0-disjoint union of its connected components (adjoined with a zero) with at least two elements.

ON THE POSET OF NON-TRIVIAL PROPER SUBGROUPS OF A FINITE GROUP

2003 ◽  
Vol 02 (02) ◽  
pp. 165-168 ◽  
Author(s):  
MARIA SILVIA LUCIDO
Keyword(s):  
Finite Group ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Connected Components ◽  
Partially Ordered ◽  
Simple Connectivity ◽  
A Finite Group

In this paper we describe the connected components of ℒ(G), the partially ordered set of non-trivial proper subgroups of a finite group G. This result is related to the study of the simple connectivity of the coset poset of a finite group.

scholarly journals On defining groups efficiently without using inverses

2002 ◽  
Vol 133 (1) ◽  
pp. 31-36 ◽  
Author(s):  
C. M. CAMPBELL ◽  
J. D. MITCHELL ◽  
N. RUšKUC
Keyword(s):  
Finite Group ◽  
Finite Semigroup ◽  
Group Presentation ◽  
Generating Set ◽  
Semigroup Presentation ◽  
A Finite Group

Let G be a group, and let 〈A[mid ]R〉 be a finite group presentation for G with [mid ]R[mid ][ges ][mid ]A[mid ]. Then there exists a finite semigroup presentation 〈B[mid ]Q〉 for G such that [mid ]Q[mid ]- [mid ]B[mid ] = [mid ]R[mid ]- [mid ]A[mid ]. Moreover, B is either the same generating set or else it contains one additional generator.

Groups whose bipartite divisor graph for character degrees has four or fewer vertices

2018 ◽  
Vol 13 (02) ◽  
pp. 2050039
Author(s):  
S. A. Moosavi
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Disjoint Union ◽  
Character Degrees ◽  
A Finite Group ◽  
Nonlinear Irreducible Character

Let [Formula: see text] be a finite group and [Formula: see text] be the set of nonlinear irreducible character degrees of [Formula: see text]. Suppose that [Formula: see text] is the set of primes dividing some elements of [Formula: see text]. The bipartite divisor graph for [Formula: see text], [Formula: see text], is a graph whose vertices are the disjoint union of [Formula: see text] and [Formula: see text], and a vertex [Formula: see text] is connected to a vertex [Formula: see text] if and only if [Formula: see text]. In this paper, we consider groups whose graph has four or fewer vertices. We show that all these groups are solvable and determine the structure of these groups. We also provide examples of any possible graph.

scholarly journals CONNECTED COMPONENTS IN THE INVARIABLY GENERATING GRAPH OF A FINITE GROUP

2021 ◽  
pp. 1-11
Author(s):  
DANIELE GARZONI
Keyword(s):  
Finite Group ◽  
Connected Components ◽  
Generating Graph ◽  
A Finite Group

Abstract We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.

The cyclic hypergroup associated with Sn, its automorphism group and its fuzzy grade

2018 ◽  
Vol 10 (05) ◽  
pp. 1850070
Author(s):  
M. Al Tahan ◽  
B. Davvaz
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Symmetric Group ◽  
Algebraic Structure ◽  
Function Composition ◽  
Finite Period ◽  
A Finite Group ◽  
Single Power

Hyperstructure is the generalized concept of algebraic structure. The study of it and its applications is extremely interesting, offering interesting results to one who is willing to look for structure. This paper deals with hyperstructures associated to the symmetric group [Formula: see text]. First, we define a new hyperoperation [Formula: see text] on [Formula: see text] and study its properties. Next, we prove that [Formula: see text] is a single-power cyclic hypergroup with finite period. Then, we determine the set of all automorphisms of [Formula: see text] and prove that it is a finite group under the operation of function composition. Finally, we construct a sequence of join spaces on [Formula: see text] and find its fuzzy grade.

scholarly journals On Thompson’s Conjecture for Alternating GroupsAp+3

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shitian Liu ◽  
Yong Yang
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Conjugacy Classes ◽  
Connected Components ◽  
Prime Divisors ◽  
Vertex Set ◽  
Thompson’S Conjecture ◽  
Trivial Center ◽  
A Finite Group

LetGbe a group. Denote byπ(G)the set of prime divisors of|G|. LetGK(G)be the graph with vertex setπ(G)such that two primespandqinπ(G)are joined by an edge ifGhas an element of orderp·q. We sets(G)to denote the number of connected components of the prime graphGK(G). Denote byN(G)the set of nonidentity orders of conjugacy classes of elements inG. Alavi and Daneshkhah proved that the groups,Anwheren=p,p+1,p+2withs(G)≥2, are characterized byN(G). As a development of these topics, we will prove that ifGis a finite group with trivial center andN(G)=N(Ap+3)withp+2composite, thenGis isomorphic toAp+3.

scholarly journals Hurwitz components of groups with socle PSL(3, q)

2021 ◽  
Vol 36 (1) ◽  
pp. 51-62
Author(s):  
H.M. Mohammed Salih
Keyword(s):  
Finite Group ◽  
Monodromy Group ◽  
Riemann Sphere ◽  
Complete List ◽  
Simple Groups ◽  
Connected Components ◽  
Branch Points ◽  
Almost Simple Groups ◽  
Hurwitz Space ◽  
A Finite Group

For a finite group G, the Hurwitz space Hinr,g(G) is the space of genus g covers of the Riemann sphere P1 with r branch points and the monodromy group G. In this paper, we give a complete list of some almost simple groups of Lie rank two. That is, we assume that G is a primitive almost simple groups of Lie rank two. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in Hinr,g(G).

scholarly journals Lower bounds for the radical of the group algebra of a finite p-soluble group

1968 ◽  
Vol 16 (2) ◽  
pp. 127-134 ◽  
Author(s):  
D. A. R. Wallace
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Lower Bounds ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Soluble Group ◽  
Nilpotent Ideal ◽  
A Finite Group ◽  
Radical Form

Over a field of characteristic p>0 the group algebra of a finite group has a unique maximal nilpotent ideal, the Jacobson radical of the algebra. The powers of the radical form a decreasing and ultimately vanishing series of ideals and it would be of interest to determine the least vanishing power. Apart from the work of Jennings (3) on p-groups little is known in general (cf. (5)) about this particular power of the radical (cf. Remarks of Brauer in (4), p. 144. Problem 15). In this paper we give non-trivial lower bounds for the index of the least vanishing power of the radical when the group is p-soluble. Of the lower bounds we give we show that that lower bound, which is dependent solely on the order of the group, is the best possible such bound and that this bound is invalid if the assumption of p-solubility is omitted.

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