The primitive conjugacy classes and the first homology group of a finite graph

The distribution of primitive conjugacy classes of the fundamental group of graphs in the first homology group of the graphs is studied in some special cases. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 154–165. Translated by A. M. Nikitin.     

On a graph associated to invariant conjugacy classes of finite groups

LetA andG be finite groups of coprime orders such thatA acts by automorphisms onG. We define theA-invariant conjugacy class graph ofG to be the graph having as vertices the noncentralA-invariant conjugacy classes ofG, and two vertices are connected by an edge if their cardinalities are not coprime. We prove that when the graph is disconnected thenG is solvable.     

Applications of a graph related to conjugacy classes in finite groups

This article was written by the authors with a during the forth author's visit to Milano and, independently.     

The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.     

On the number of conjugacy classes in a finite group II

In this work we obtain new properties connected with the number of conjugacy classes of elements of a finite group, through the analysis of the numberr _G(gN) of conjugacy classes of elements ofG that intersect the cosetgN, whereN is a normal subgroup ofG andg any element ofG. The results obtained about this number are not only used in the general problem of classifying finite groups according to the number of conjugacy classes, but they also allow us to improve and generalize known results relating to conjugacy classes due to P. Hall, M. Cartwright, A. Mann, G. Sherman, A. Vera-López and L. Ortíz de Elguea. Examples are given which illustrate our improvements. This work has been supported by the University of the Basque Country.     

On the power graph and the reduced power graph of a finite group

In this paper, for a finite group, we investigate to what extent its directed (resp. undirected) reduced power graph determines its directed power graph (resp. reduced power graph). Moreover, we investigate the determination of the orders of the elements of a finite group from its directed (resp. undirected) reduced power graph. Consequently, we show that some classes of finite groups are recognizable from their undirected reduced power graphs. Also, we study the relationship between the isomorphism classes of groups corresponding to the equivalence relations induced by the isomorphism of each of these graphs on the set of all finite groups.     

On the number of conjugacy classes of maximal subgroups in a finite soluble group

We show that for many formations \frak F\frak F, there exists an integer n = [`(m)](\frak F)n = \overline m(\frak F) such that every finite soluble group G not belonging to the class \frak F\frak F has at most n conjugacy classes of maximal subgroups belonging to the class \frak F\frak F. If \frak F\frak F is a local formation with formation function f, we bound [`(m)](\frak F)\overline m(\frak F) in terms of the [`(m)](f(p))(p ? \Bbb P )\overline m(f(p))(p \in \Bbb P ). In particular, we show that [`(m)](\frak N^k) = k+1\overline m(\frak N^k) = k+1 for every nonnegative integer k, where \frak N^k\frak N^k is the class of all finite groups of Fitting length ￡ k\le k.     

The monomiality conditions for a finite group with few conjugacy classes lying outside a normal subgroup

A finite group with a normal subgroup containing almost all conjugacy classes except one, two, or three ones is considered. Sufficient monomiality conditions for such group are found for each of these cases. It turns out that additional conditions are not required in the first case.     

Lambda number of the power graph of a finite group

Journal of Algebraic Combinatorics - The power graph $$\Gamma _G$$ of a finite group G is the graph with the vertex set G, where two distinct elements are adjacent if and only if one is a power of...     

A note on conjugacy classes of finite groups

Let G be a finite group and let x ^G denote the conjugacy class of an element x of G. We classify all finite groups G in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G.     

On C-small conjugacy classes in a reductive group

Let G be an almost simple reductive group with Weyl group W. Let B be a Borel subgroup of G. Let w be an elliptic element of W which has minimal length in its conjugacy class. We show that in almost all cases there exists a semisimple class in G whose intersection with BwB has dimension dim(B).     

Arithmetical conditions on the conjugacy vector of a finite group

We get new properties of the numbersr _G(xN) = |{Cl _G (g)|Cl _G (g)∩xN ≠ Ø} (whereG is a finite group andN is a normal subgroup ofG) that are useful in the analysis of the classification of the finite groups according to the number of conjugacy classes.     

Notes on the length of conjugacy classes of finite groups

The purpose of this paper is to investigate influences of lengths of conjugacy classes of finite groups on the structure of finite groups. We get a necessary and sufficient condition for a finite group G to be equal to O_p(G)×O_p^′(G). We also generalize some results (Comm. Algebra 27 (9) (1999) 4347).     

Remarks on the length of conjugacy classes of finite groups

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p ² for some prime p. We generalise and provide simplified proofs for some earlier results.     

Triangles in the graph of conjugacy classes of normal subgroups

Let G be a finite group and N a normal subgroup of G. We determine the structure of N when the graph \(\Gamma _{G}(N)\), which is the graph associated to the conjugacy classes of G contained in N, has no triangles and when the graph consists in exactly one triangle.     

Some finite groups with large conjugacy classes

We derive some properties of a family of finite groups, which was investigated by Camina, Macdonald, and others. For instance, we give information about the Schur multipliers of the class twop-groups in this family. A large part of this paper was written while the author was visiting the Department of Mathematics of the University of Trento. The author is indebted to this department, and in particular to C.M. Scoppola, for their kind hospitality. The author is also grateful to D. Chillag for his constructively destructive criticism of the first version of this paper.     

On the order supergraph of the power graph of a finite group

Ricerche di Matematica - The power graph $${\mathcal {P}}_{G}$$ of a finite group G is the graph whose vertex set is G, two distinct vertices are adjacent if one is a power of the other. The order...     

On the independence number of the power graph of a finite group

The power graph ${\Gamma }_G$ of a finite group $G$ is the graph whose vertex set is $G$, two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of ${\Gamma }_G$ and characterize the groups achieving the bounds. Moreover, we determine the independence number of ${\Gamma }_G$ if $G$ is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups $G$ whose power graphs have independence number 3 or $n?2$, where $n$ is the order of $G$.