On decomposability of finite groups

Let G be a finite group. A normal subgroup N of G is a union of several G-conjugacy classes, and it is called n-decomposable in G if it is a union of n distinct G-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.     

Groups with some arithmetic conditions on real class sizes

Let G be a finite group. An x∈G is a real element if x and x ^?1 are conjugate in G. For x∈G, the conjugacy class x ^G is said to be a real conjugacy class if every element of x ^G is real. We show that if 4 divides no real conjugacy class sizes of a finite group G, then G is solvable. We also study the structure of such groups in detail. This generalizes several results in the literature.     

The Conjugacy Class Number k(G): A Different Perspective

For a finite group G, let m(G) denote the least positive integer such that the union of any m(G) distinct nontrivial conjugacy classes of G together with the identity of G is a subgroup of G. We prove that m(G) = k(G) ?1 for all m(G) ≥2.     

On finite x-decomposable groups for X = {1, 2, 4}

A normal subgroup N of a finite group G is called an n-decomposable subgroup if N is a union of n distinct conjugacy classes of G. Each finite nonabelian nonperfect group is proved to be isomorphic to Q ₁₂, or Z ₂ × A ₄, or G = ??a, b, c | a ¹¹ = b ⁵ = c ² = 1, b ^?1 ab = a ⁴, c ^?1 ac = a ^?1, c ^?1 bc = b ^?1?? if every nontrivial normal subgroup is 2- or 4-decomposable.     

Groups with Bounded Chernikov Conjugate Classes of Elements

We consider BCC-groups, that is groups G with Chernikov conjugacy classes in which for every element x G the minimax rank of the divisible part of the Chernikov group G/C _G(x ^G) and the order of the corresponding factor-group are bounded in terms of G only. We prove that a BCC-group has a Chernikov derived subgroup. This fact extends the well-known result due to B. H. Neumann characterizing groups with bounded finite conjugacy classes (BFC-groups).     

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.     

Derived length and conjugacy class sizes

Let G be a finite solvable group, and let F(G) be its Fitting subgroup. We prove that there is a universal bound for the derived length of G/F(G) in terms of the number of distinct conjugacy class sizes of G. This result is asymptotically best possible. It is based on the following result on orbit sizes in finite linear group actions: If G is a finite solvable group and V a finite faithful irreducible G-module of characteristic r, then there is a universal logarithmic bound for the derived length of G in terms of the number of distinct r^′-parts of the orbit sizes of G on V. This is a refinement of the author's previous work on orbit sizes.     

On a graph related to permutability in finite groups

For a finite group G we define the graph Γ(G) to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes ${\mathcal {A}, \mathcal {B}}For a finite group G we define the graph Γ(G) to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes A, B{\mathcal {A}, \mathcal {B}} are joined by an edge if for some A ? A, B ? B A{A \in \mathcal {A},\, B \in \mathcal {B}\, A} and B permute. We characterise those groups G for which Γ(G) is complete.     

A description of Baer-Suzuki type of the solvable radical of a finite group

We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g∈G such that for any 3 elements a₁,a₂,a₃∈G the subgroup generated by the elements , i=1,2,3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2,3}^′-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).     

On a Commuting Graph on Conjugacy Classes of Groups

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.     

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.     

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〉.     

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 9- and 10-decomposable finite groups

A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a union of n distinct conjugacy classes of G. In some research papers, the question of finding all positive integer n such that there is an n-decomposable finite group was posed. In this paper, we investigate the structure of 9- and 10-decomposable non-perfect finite groups. We prove that a non-perfect group G is 9-decomposable if and only if G is isomorphic to Aut(PSL(2,32)), Aut(PSL(3,3)), the semi-direct product Z ₃ ∝(Z ₅×Z ₅) or a non-abelian group of order pq, where p and q are primes and p?1=8q, and also, a non-perfect finite group G is 10-decomposable if and only if G is isomorphic to Aut(PSL(2,17)), PSL(2,25):2₃, a split extension of PSL(2,25) by Z ₂ in ATLAS notation (Conway et al., Atlas of Finite Groups, [1985]), Aut(U ₃(3)) or D ₃₈, where D ₃₈ denotes the dihedral group of order 38.     

Conjugacy classes and finite p-groups

Let G be a finite p-group, where p is a prime number, and a ∈ G. Denote by Cl(a) = {gag⁻¹| g ∈ G} the conjugacy class of a in G. Assume that |Cl(a)| = pⁿ. Then Cl(a) Cl(a⁻¹) = {xy | x ∈ Cl(a), y ∈ Cl(a⁻¹)} is the union of at least n(p − 1) + 1 distinct conjugacy classes of G. Received: 16 December 2004     

Partially harmonic spinors and representations of reductive Lie groups

Let G be a reductive Lie group subject to some minor technical restrictions. The Plancherel Theorem for G uses several series of unitary representation classes, one series for each conjugacy class of Cartan subgroups of G. Given a Cartan subgroup H ? G, we construct a G-homogeneous family X → Y of oriented riemannian symmetric spaces, some G-homogeneous bundles , and some Hilbert spaces of partially harmonic spinors with values in . Then G acts on by a unitary representation π_μ,σ^±. We then show that these π_μ,σ^± realize the series of representation classes of G associated to the conjugacy class of H.     

Conjugacy classes of derangements in finite transitive groups

Let G be a permutation group acting transitively on a finite set Ω. We classify all such (G, Ω) when G contains a single conjugacy class of derangements. This was done under the assumption that G acts primitively by Burness and Tong-Viet. It turns out that there are no imprimitive examples. We also discuss some results on the proportion of conjugacy classes which consist of derangements.     

Residually finite algorithmically finite groups,their subgroups and direct products

We construct a finitely generated infinite recursively presented residually finite algorithmically finite group G, thus answering a question of Myasnikov and Osin. The group G here is “strongly infinite” and “strongly algorithmically finite,” which means that G contains an infinite Abelian normal subgroup and all finite Cartesian powers of G are algorithmically finite (i.e., for any n, there is no algorithm writing out infinitely many pairwise distinct elements of the group G ⁿ ). We also formulate several open questions concerning this topic.     

A characterization of fuzzy subgroups

An ordinary subgroup of a group G is (1) a subset of G, (2) closed under the group operation. In a fuzzy subgroup it is precisely these two notions that lose their deterministic character. A fuzzy subgroup μ of a group (G,·) associates with each group element a number, the larger the number the more certainly that element belongs to the fuzzy subgroup. The closure property is captured by the inequality μ(x · y)?T(μ(x), μ(y)). In A. Rosenfeld's original definition, T was the function ‘minimum’. However, any t-norm T provides a meaningful generalization of the closure property. Two classes of fuzzy subgroups are investigated. The fuzzy subgroups in one class are subgroup generated, those in the other are function generated. Each fuzzy subgroup in these classes satisfies the above inequality with T given by T(a, b) = max(a + b ?1, 0). While the two classes look different, each fuzzy subgroup in either is isomorphic to one in the other. It is shown that a fuzzy subgroup satisfies the above inequality with $\text{T =}\text{‘minimum’}$ if and only if it is subgroup generated of a very special type. Finally, these notions are applied to some abstract pattern recognition problems.     

Hyperbolic elements in negatively curved groups

We show that in a negatively curved groupG the conjugacy class of any infinite cyclic subgroup contains a straight element, an elementg with |g ⁿ |=n|g|, and thus the translation number of an element in a negatively curved group is rational with uniformly bounded denominator. We also find an upper bound on the cardinality of a finite normal subgroup.