scholarly journals The genus of the subgroup graph of a finite group

2020 ◽  
pp. 2050010
Author(s):  
Andrea Lucchini
Keyword(s):  
Finite Group ◽  
Prime Divisors ◽  
A Finite Group ◽  
Number Of Prime Divisors

For a finite group [Formula: see text] denote by [Formula: see text] the genus of the subgroup graph of [Formula: see text] We prove that [Formula: see text] tends to infinity as either the rank of [Formula: see text] or the number of prime divisors of [Formula: see text] tends to infinity.

A generalization of Hall’s theorem

2016 ◽  
Vol 10 (02) ◽  
pp. 1750024
Author(s):  
Feng Zhou
Keyword(s):  
Finite Group ◽  
Prime Divisors ◽  
Hall's Theorem ◽  
A Finite Group ◽  
Hall’S Theorem

Let [Formula: see text] be a finite group, whose order has [Formula: see text] prime divisors. In this paper, we prove that if [Formula: see text] has a [Formula: see text]-complement for [Formula: see text] prime divisors [Formula: see text] of [Formula: see text] and [Formula: see text] has no section isomorphic to [Formula: see text]. Then [Formula: see text] is solvable, which generalizes a theorem of Hall.

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].

scholarly journals MULTIPLICITIES IN SYLOW SEQUENCES AND THE SOLVABLE RADICAL

2008 ◽  
Vol 78 (3) ◽  
pp. 477-486 ◽  
Author(s):  
GIL KAPLAN ◽  
DAN LEVY
Keyword(s):  
Finite Group ◽  
Composition Factor ◽  
Prime Divisors ◽  
Solvable Radical ◽  
A Finite Group

AbstractA complete Sylow sequence, 𝒫=P1,…,Pm, of a finite group G is a sequence of m Sylow pi-subgroups of G, one for each pi, where p1,…,pm are all of the distinct prime divisors of |G|. A product of the form P1⋯Pm is called a complete Sylow product of G. We prove that the solvable radical of G equals the intersection of all complete Sylow products of G if, for every composition factor S of G, and for every ordering of the prime divisors of |S|, there exist a complete Sylow sequence 𝒫 of S, and g∈S such that g is uniquely factorizable in 𝒫 . This generalizes our results in Kaplan and Levy [‘The solvable radical of Sylow factorizable groups’, Arch. Math.85(6) (2005), 490–496].

Some sufficient conditions on solvability of finite groups

2016 ◽  
Vol 15 (03) ◽  
pp. 1650057 ◽  
Author(s):  
Wei Meng ◽  
Jiakuan Lu ◽  
Li Ma ◽  
Wanqing Ma
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Conjugacy Classes ◽  
Maximal Subgroups ◽  
Prime Divisors ◽  
Abelian Subgroups ◽  
A Finite Group

For a finite group [Formula: see text], the symbol [Formula: see text] denotes the set of the prime divisors of [Formula: see text] denotes the number of conjugacy classes of maximal subgroups of [Formula: see text]. Let [Formula: see text] denote the number of conjugacy classes of non-abelian subgroups of [Formula: see text] and [Formula: see text] denote the number of conjugacy classes of all non-normal non-abelian subgroups of [Formula: see text]. In this paper, we consider the finite groups with [Formula: see text] or [Formula: see text]. We show these groups are solvable.

scholarly journals Quasirecognition by Prime Graph of the Groups 2D2n(q) Where q < 105

2017 ◽  
Author(s):  
Hossein Moradi ◽  
Mohammad Reza Darafsheh ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Power ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
A Finite Group

Let G be a finite group. The prime graph &Gamma;(G) of G is defined as follows: The set of vertices of&nbsp;&Gamma;(G) is the set of prime divisors of |G| and two distinct vertices p and p' are connected in &Gamma;(G), whenever G has an element of order pp'. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with &Gamma;(G)=&Gamma;(P), G has a composition factor isomorphic to P. In&nbsp;[4] proved finite simple groups 2Dn(q), where&nbsp;n&nbsp;&ne; 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2D2k(q), where&nbsp;k &ge; 9 and q is a prime power less than 105.

N-critical graph of finite groups

2021 ◽  
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Group ◽  
Directed Graph ◽  
Finite Groups ◽  
Soluble Group ◽  
Prime Divisors ◽  
Nilpotent Length ◽  
Critical Graph ◽  
Vertex Set ◽  
A Finite Group

A Schmidt [Formula: see text]-group is a non-nilpotent [Formula: see text]-group whose proper subgroups are nilpotent and which has the normal Sylow [Formula: see text]-subgroup. The [Formula: see text]-critical graph [Formula: see text] of a finite group [Formula: see text] is a directed graph on the vertex set [Formula: see text] of all prime divisors of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] has a Schmidt [Formula: see text]-subgroup. The bounds of the nilpotent length of a soluble group are obtained in terms of its [Formula: see text]-critical graph. The structure of a soluble group with given [Formula: see text]-critical graph is obtained in terms of commutators. The connections between [Formula: see text]-critical and other graphs (Sylow, soluble, prime, commuting) of finite groups are found.

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.

On Prime Spectrum of Maximal Subgroups in Finite Groups

2018 ◽  
Vol 25 (04) ◽  
pp. 579-584
Author(s):  
Chi Zhang ◽  
Wenbin Guo ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Prime Spectrum ◽  
Prime Divisors ◽  
A Finite Group ◽  
Theoretical Results

For a positive integer n, we denote by π(n) the set of all prime divisors of n. For a finite group G, the set [Formula: see text] is called the prime spectrum of G. Let [Formula: see text] mean that M is a maximal subgroup of G. We put [Formula: see text] and [Formula: see text]. In this notice, using well-known number-theoretical results, we present a number of examples to show that both K(G) and k(G) are unbounded in general. This implies that the problem “Are k(G) and K(G) bounded by some constant k?”, raised by Monakhov and Skiba in 2016, is solved in the negative.

scholarly journals Two problems on finite groups with k conjugate classes

1968 ◽  
Vol 8 (1) ◽  
pp. 49-55 ◽  
Author(s):  
John Poland
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Prime Divisors ◽  
Image Position ◽  
A Finite Group ◽  
Generally True

Let G be a finite group of order g having exactly k conjugate classes. Let π(G) denote the set of prime divisors of g. K. A. Hirsch [4] has shown that By the same methods we prove g ≡ k modulo G.C.D. {(p–1)2 p ∈ π(G)} and that if G is a p-group, g = h modulo (p−1)(p2−1). It follows that k has the form (n+r(p−1)) (p2−1)+pe where r and n are integers ≧ 0, p is a prime, e is 0 or 1, and g = p2n+e. This has been established using representation theory by Philip Hall [3] (see also [5]). If then simple examples show (for 6 ∤ g obviously) that g ≡ k modulo σ or even σ/2 is not generally true.

Lower bounds on conjugacy classes of non-nilpotent subgroups in a finite group

2014 ◽  
Vol 14 (03) ◽  
pp. 1550039 ◽  
Author(s):  
Wei Meng ◽  
Jiakuan Lu
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Lower Bounds ◽  
Conjugacy Classes ◽  
Finite Solvable Group ◽  
Prime Divisors ◽  
A Finite Group

For a finite group G, let γ(G) denote the number of conjugacy classes of all non-nilpotent subgroups of G, and let π(G) denote the set of the prime divisors of |G|. In this paper, we establish lower bounds on γ(G). In fact, we show that if G is a finite solvable group, then γ(G) = 0 or γ(G) ≥ 2|π(G)|-2, and if G is non-solvable, then γ(G) ≥ |π(G)| + 1. Both lower bounds are best possible.

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