The power graph on the conjugacy classes of a finite group

A digraph \({\overrightarrow{\mathcal{Pc}}(G)}\) is said to be the directed power graph on the conjugacy classes of a group G, if its vertices are the non-trivial conjugacy classes of G, and there is an arc from vertex C to C′ if and only if \({C \neq C'}\) and \({C \subseteqq {C'}^{m}}\) for some positive integer \({m > 0}\). Moreover, the simple graph \({\mathcal{Pc}(G)}\) is said to be the (undirected) power graph on the conjugacy classes of a group G if its vertices are the conjugacy classes of G and two distinct vertices C and C′ are adjacent in \({\mathcal{Pc}(G)}\) if one is a subset of a power of the other. In this paper, we find some connections between algebraic properties of some groups and properties of the associated graph.     

On the largest conjugacy class length of a finite group

Let $G$ be a finite group and $\mathrm{bcl}(G)$ the largest conjugacy class length of $G$ . In this note we slightly improve He and Shi’s lower bound for $\mathrm{bcl}(G)$ , showing that $|\mathrm{bcl}(G)|\ge p^{\frac{1}{p}}(|G:O_{p}(G)|_{p})^{\frac{p-1}{p}}$ .     

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.     

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.     

Multiplying a conjugacy class by its inverse in a finite group

Suppose that G is a finite group and K is a non-trivial conjugacy class of G such that KK^?1 = 1 ∪ D ∪ D^?1 with D a conjugacy class of G. We prove that G is not a non-abelian simple group and we give arithmetical conditions on the class sizes determining the solvability and the structure of 〈K〉 and 〈D〉.     

Monomiality of finite groups with some conditions on conjugacy classes

We present some arithmetical-type conditions on the set of conjugacy classes of a finite group that are sufficient for the monomiality of the group, i.e., for the property that all its irreducible complex characters are induced by linear characters of subgroups. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 201–212, 2007.     

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 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.     

On conjugacy class sizes of primary and biprimary elements of a finite group

Let m, n > 1 be two coprime integers. In this paper, we prove that a finite solvable group is nilpotent if the set of the conjugacy class sizes of its primary and biprimary elements is {1,m, n,mn}.     

On stable conjugacy of finite subgroups of the plane Cremona group, I

We discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We compute the stable birational invariant H ¹(G, Pic(X)) for cyclic groups of prime order.     

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials and , with nonnegative integral coefficients, such that the vector bundle is isomorphic to . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group.

When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

     

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.     

Commuting elements in a conjugacy class of finite groups

We presentmethods for verification of the following conjecture: A nonidentity conjugacy class in a finite simple group contains two commuting elements. By way of illustration, we consider sporadic groups, the projective group L_n(q) and the alternating group A_n.     

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.     

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).