N-critical graph of finite groups

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.

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

10.1142/s0219498821500444 ◽

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 ◽

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].

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

Open Mathematics ◽

10.1515/math-2021-0071 ◽

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 ◽

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.

Proper connection of power graphs of finite groups

10.1142/s021949882150033x ◽

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 ◽

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].

Some sufficient conditions on solvability of finite groups

10.1142/s0219498816500572 ◽

2016 ◽

Vol 15 (03) ◽

pp. 1650057 ◽

Cited By ~ 1

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 ◽

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.

Automorphisms of Finite Groups and their Fixed-Point Groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700007424 ◽

1969 ◽

Vol 9 (3-4) ◽

pp. 467-477 ◽

Cited By ~ 11

Author(s):

J. N. Ward

Keyword(s):

Fixed Point ◽

Finite Group ◽

Abelian Group ◽

Finite Groups ◽

Upper Bound ◽

Prime Order ◽

Soluble Group ◽

Point Groups ◽

Derived Series ◽

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.

Non-Abelian finite groups whose character sums are invariant but are not Cayley isomorphism

10.1142/s0219498819500130 ◽

2019 ◽

Vol 18 (01) ◽

pp. 1950013

Author(s):

Alireza Abdollahi ◽

Maysam Zallaghi

Keyword(s):

Finite Group ◽

Cayley Graph ◽

Finite Groups ◽

Closed Subset ◽

Cayley Graphs ◽

Character Sums ◽

Main Question ◽

Vertex Set ◽

A Finite Group ◽

Positive Integers

Let [Formula: see text] be a group and [Formula: see text] an inverse closed subset of [Formula: see text]. By a Cayley graph [Formula: see text], we mean the graph whose vertex set is the set of elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. A group [Formula: see text] is called a CI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for some automorphism [Formula: see text] of [Formula: see text]. A finite group [Formula: see text] is called a BI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for all positive integers [Formula: see text], where [Formula: see text] denotes the set [Formula: see text]. It was asked by László Babai [Spectra of Cayley graphs, J. Combin. Theory Ser. B 27 (1979) 180–189] if every finite group is a BI-group; various examples of finite non-BI-groups are presented in [A. Abdollahi and M. Zallaghi, Character sums of Cayley graph, Comm. Algebra 43(12) (2015) 5159–5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However, it is known that there are finite abelian non-CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders 20 and 42 which are not CI-groups. We also list all BI-groups of orders up to 30.

Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups

Filomat ◽

10.2298/fil1716323h ◽

2017 ◽

Vol 31 (16) ◽

pp. 5323-5334 ◽

Cited By ~ 1

Author(s):

Asma Hamzeh ◽

Ali Ashrafi

Keyword(s):

Finite Group ◽

Characteristic Polynomial ◽

Finite Groups ◽

The Other ◽

Laplacian Spectrum ◽

Simple Graphs ◽

Power Graph ◽

Vertex Set ◽

Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x,y ? G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.

Some results on the main supergraph of finite groups

Algebra and Discrete Mathematics ◽

10.12958/adm584 ◽

2020 ◽

Vol 30 (2) ◽

pp. 172-178

Author(s):

A. K. Asboei ◽

◽

S. S. Salehi ◽

Keyword(s):

Finite Group ◽

Simple Group ◽

Finite Groups ◽

Sporadic Simple Group ◽

Vertex Set ◽

Let G be a finite group. The main supergraph S(G) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o(x)∣o(y) or o(y)∣o(x). In this paper, we will show that G≅PSL(2,p) or PGL(2,p) if and only if S(G)≅S(PSL(2,p)) or S(PGL(2,p)), respectively. Also, we will show that if M is a sporadic simple group, then G≅M if only if S(G)≅S(M).

Formations of Π-soluble groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700007138 ◽

1969 ◽

Vol 10 (1-2) ◽

pp. 241-250 ◽

Cited By ~ 6

Author(s):

H. Lausch

Keyword(s):

Finite Group ◽

Finite Groups ◽

Soluble Group ◽

Composition Series ◽

Soluble Groups ◽

Hall Subgroups ◽

The theory of formations of soluble groups, developed by Gaschütz [4], Carter and Hawkes[1], provides fairly general methods for investigating canonical full conjugate sets of subgroups in finite, soluble groups. Those methods, however, cannot be applied to the class of all finite groups, since strong use was made of the Theorem of Galois on primitive soluble groups. Nevertheless, there is a possiblity to extend the results of the above mentioned papers to the case of Π-soluble groups as defined by Čunihin [2]. A finite group G is called Π-soluble, if, for a given set it of primes, the indices of a composition series of G are either primes belonging to It or they are not divisible by any prime of Π In this paper, we shall frequently use the following result of Čunihin [2]: Ift is a non-empty set of primes, Π′ its complement in the set of all primes, and G is a Π-soluble group, then there always exist Hall Π-subgroups and Hall ′-subgroups, constituting single conjugate sets of subgroups of G respectively, each It-subgroup of G contained in a Hall Π-subgroup of G where each ′-subgroup of G is contained in a Hall Π′-subgroup of G. All groups considered in this paper are assumed to be finite and Π-soluble. A Hall Π-subgroup of a group G will be denoted by G.

On Thompson’s Conjecture for Alternating GroupsAp+3

The Scientific World JOURNAL ◽

10.1155/2014/752598 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 1

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 ◽

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.