scholarly journals Regularity of extended conjugate graphs of finite groups

2022 ◽  
Vol 7 (4) ◽  
pp. 5480-5498
Author(s):  
Piyapat Dangpat ◽  
◽  
Teerapong Suksumran ◽  
Keyword(s):  
Finite Groups ◽  
Necessary Conditions ◽  
Undirected Graph ◽  
Algebraic Property ◽  
Sufficient Conditions ◽  
Strong Connection ◽  
Theoretic Property ◽  
Graph Theoretic ◽  
Vertex Set ◽  
A Finite Group

<abstract><p>The extended conjugate graph associated to a finite group $ G $ is defined as an undirected graph with vertex set $ G $ such that two distinct vertices joined by an edge if they are conjugate. In this article, we show that several properties of finite groups can be expressed in terms of properties of their extended conjugate graphs. In particular, we show that there is a strong connection between a graph-theoretic property, namely regularity, and an algebraic property, namely nilpotency. We then give some sufficient conditions and necessary conditions for the non-central part of an extended conjugate graph to be regular. Finally, we study extended conjugate graphs associated to groups of order $ pq $, $ p^3 $, and $ p^4 $, where $ p $ and $ q $ are distinct primes.</p></abstract>

Proper connection of power graphs of finite groups

2020 ◽  
pp. 2150033
Author(s):  
Xuanlong Ma
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Undirected Graph ◽  
The Other ◽  
Connection Number ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Connection Number ◽  
A Finite Group

Let [Formula: see text] be a finite group. The power graph of [Formula: see text] is the undirected graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of [Formula: see text] is the subgraph of the power graph of [Formula: see text] obtained by deleting all edges [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are two distinct elements of [Formula: see text]. In this paper, we determine the proper connection number of the reduced power graph of [Formula: see text]. As an application, we also determine the proper connection number of the power graph of [Formula: see text].

scholarly journals Certain properties of the power graph associated with a finite group

2014 ◽  
Vol 13 (07) ◽  
pp. 1450040 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
S. Rahbariyan ◽  
W. J. Shi
Keyword(s):  
Finite Groups ◽  
Identity Element ◽  
Sufficient Conditions ◽  
Strongly Regular Graph ◽  
Simple Graph ◽  
Power Graph ◽  
Vertex Set ◽  
Strongly Regular ◽  
A Finite Group ◽  
Necessary And Sufficient

The power graph [Formula: see text] of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if one of them is a power of the other. The subgraph [Formula: see text] of [Formula: see text] is obtained by deleting the vertex 1 (the identity element of G). In this paper, we first investigate some properties of the power graph [Formula: see text] and its subgraph [Formula: see text]. We next provide necessary and sufficient conditions for a power graph [Formula: see text] to be a strongly regular graph, a bipartite graph or a planar graph. Finally, we obtain some infinite families of finite groups G for which the power graph [Formula: see text] contains some cut-edges.

Finite groups with two supersoluble subgroups

2019 ◽  
Vol 22 (2) ◽  
pp. 297-312 ◽  
Author(s):  
Victor S. Monakhov ◽  
Alexander A. Trofimuk
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Derived Subgroup ◽  
Nilpotent Residual ◽  
A Finite Group

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

scholarly journals Finite groups whose intersection power graphs are toroidal and projective-planar

2021 ◽  
Vol 19 (1) ◽  
pp. 850-862
Author(s):  
Huani Li ◽  
Xuanlong Ma ◽  
Ruiqin Fu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Identity Element ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

Abstract The intersection power graph of a finite group G G is the graph whose vertex set is G G , and two distinct vertices x x and y y are adjacent if either one of x x and y y is the identity element of G G , or ⟨ x ⟩ ∩ ⟨ y ⟩ \langle x\rangle \cap \langle y\rangle is non-trivial. In this paper, we completely classify all finite groups whose intersection power graphs are toroidal and projective-planar.

The Influence of the Number of Non-(sub)normal Non-abelian Subgroups on Solvability of Finite Groups

2016 ◽  
Vol 23 (02) ◽  
pp. 325-328
Author(s):  
Jiangtao Shi ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Abelian Subgroups ◽  
A Finite Group

We obtain some sufficient conditions on the number of non-(sub)normal non-abelian subgroups of a finite group to be solvable, which extend a result of Shi and Zhang in 2011.

On SS-supplemented subgroups of some sylow subgroups of finite groups

2021 ◽  
Author(s):  
Xuanli He ◽  
Qinghong Guo ◽  
Muhong Huang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Necessary Conditions ◽  
Saturated Formation ◽  
Sylow Subgroups ◽  
Group A ◽  
Sufficient And Necessary Conditions ◽  
A Finite Group

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] permutes with all Sylow subgroups of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-supplemented in [Formula: see text] if there exists a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate [Formula: see text]-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

scholarly journals On the minimum degree of the power graph of a finite cyclic group

2020 ◽  
pp. 2150044
Author(s):  
Ramesh Prasad Panda ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Minimum Degree ◽  
Simple Graph ◽  
The Other ◽  
Prime Divisors ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

Solubility theorems for finite groups

1973 ◽  
Vol 73 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Thomas J. Laffey
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
A Finite Group

In this paper we obtain various sufficient conditions for the solubility of a finite group. In particular, we show that if G is a finite group and p≥5 is a prime such that all p′-subgroups of G are nilpotent, then G is soluble. We show also that if G is a finite group which has a cyclic Sylow p-subgroup Pand such that for all p′-subgroups H of G, H is nilpotent and H′ is cyclic, then, if p≠3, either P◃G or G has a normal p-complement.

scholarly journals On critical groups

1966 ◽  
Vol 6 (2) ◽  
pp. 237-250 ◽  
Author(s):  
L. G. Kovács ◽  
M. F. Newman
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Minimal Normal Subgroup ◽  
Sufficient Conditions ◽  
Critical Groups ◽  
Necessary Condition ◽  
Important Paper ◽  
Trivial Group ◽  
Proper Factor ◽  
A Finite Group

The concept of critical group was introduced by D. C. Cross (as reported by G. Higman in [5]): a finite group is called critical if it is not contained in the variety generated by its proper factors. (The factors of a group G are the groups H/K where K H ≦ G, and H/K is a proper factor of G unless H = G and K =1). Some investigations concerning finite groups and varieties depend on the investigator's ability to decide whether a given group is critical or not. (For instance, one of the crucial points in the important paper [9] of Sheila Oates and M. B. Powell is a necessary condition of criticality: their Lemma 2.4.2.) An obvious necessary condition is that the group should have only one minimal normal subgroup: the group is then called monolithic, and the minimal normal subgroup its monolith. This is, however, far from being a sufficient condition, and it is the purpose of the present paper to give some sufficient conditions for the criticality of monolithic groups. (We consider the trivial group neither monolithic nor critical.) The basis of our results is an analysis of the following situation.

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