# conjugacy class sizes

## Top PDF conjugacy class sizes: ### Finite groups with the same conjugacy class sizes as a finite simple group

Proof. (i) By Lemma 2.5, | S | | | G/Z(G) | . Now let there exist q ∈ π(G/Z (G)) − π(S). Then since q ∤ | S | , we get that q does not divide any conjugacy class sizes of G and hence, Lemma 2.1 forces the q-Sylow subgroup of G to be a subgroup of Z(G), so q ̸∈ π(G/Z(G)), which is a contradiction. This yields that π(G/Z (G)) = π(S). If q ∈ π(G/Z (G)) = π(S) such that | G/Z(G) | q ̸ = | S | q , then ### On finite groups with square-free conjugacy class sizes

Only finite groups will be considered in this expository paper. Over the last decades, the influence of conjugacy class sizes on the structure of finite groups has been thoroughly investigated. For instance, a detailed account on this subject is collected in the survey of Camina and Camina . We concentrate here on a particular arithmetical property of the conjugacy class sizes: when they are square-free. Since that survey, some new progress has been made in this direction. In particular, we are interested in such arithmetical property in the context of factorised groups. We will outline many of the known results in this line and we will provide references to the literature for their proofs. ### On p regular G conjugacy class sizes

In a recent paper [], the authors studied the structure of G under the condition that the largest two p-regular conjugacy class sizes (say, m and n) of G are coprime, where m > n and p n. Notice that, when G = N, the condition n dividing |N /Z(N)| is of course true, so our aim is, by eliminating the assumption p n, to investigate the properties of N under the corresponding condition. More precisely, we prove the following. ### One-prime power hypothesis for conjugacy class sizes

Abstract. A finite group G satisfies the one-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes m and n are either equal or have common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to S × A where A is abelian and S ∼ = P SL 2 (q) for q ∈ { 4, 8 } . We confirm this conjecture. ### Conjugacy classes and graphs of two-groups of nilpotency class two

Chapter 2 presents the literature review of this research. Some basic definitions and concepts related to this research are presented. Various works by different researchers concerning the bound and the exact number of conjugacy classes, conjugacy class sizes and graphs related to conjugacy classes are compared and stated. Furthermore, the classifications of 2-generator 2-groups of class two and the application of GAP in this research are given in this chapter. ### On bipartite divisor graph for character degrees

set X ∗ and two vertices m and n are connected if gcd(m, n) > 1. Motivated by , in this paper, we introduce the bipartite divisor graph for character degrees of a finite group and obtain some properties of this graph. The concept of the bipartite divisor graph for integer subsets has been considered in  and the bipartite divisor graph for group conjugacy class sizes has been studied in  and . The bipartite divisor graph B (X) is the graph with vertex set the disjoint union 1 ρ(X) ∪ X ∗ and edge set ### Bipartite divisor graph for the set of irreducible character degrees

The notion of the bipartite divisor graph was first introduced by Iranmanesh and Praeger in  for a finite set of positive integers. As an application of this graph in group theory, in , the writers considered this graph for the set of conjugacy class sizes of a finite group and studied various properties of it. In particular they proved that the diameter of this graph is at most six, and classified those groups for which the bipartite divisor graphs of conjugacy class sizes have diameter exactly 6. Moreover, they showed that if the graph is acyclic, then the diameter is at most five and they classified the groups for which the graph is a path of length five. Similarly, Taeri in  considered the case that the bipartite divisor graph of the set of conjugacy class sizes is a cycle and (by using the structure of F -groups and the classification of finite simple groups) proved that for a finite nonabelian group G, the bipartite divisor graph of the conjugacy class sizes is a cycle if and only if it is a cycle of length 6, and for an abelian group A and q ∈ { 4, 8 } , G ≃ A × SL 2 (q). Inspired by these papers, in this work we consider ### Conjugacy and order classes of two-generator p-groups of nilpotency class two

Studies on the bounds of the number of conjugacy classes done by a number of researchers are reviewed. The classification of 2-generator p-groups of class 2 (p an odd prime) is stated and elaborated. Then, more background on the software package GAP used in this research is explained. ### The conjugacy classes of metabelian groups of order at most 24

classes as well as non-central conjugacy classes. Moreover, the metabelian groups of types (9), (10) and 12) have the same number of non-central conjugacy classes of order greater than one, namely 5, and the metabelian groups types (13), (15), (16) and (17) also have the same number of conjugacy classes which is 10 as well as non-central conjugacy classes of 6 of the same sizes. However, the metabelian groups types (14), (20), (23) have the same number of non-central conjugacy classes of 5. Whereas, the metabelian groups of types (19), (21) and (25) each has a unique number of conjugacy classes of 9, 8 and 7 respectively. ### Recognizing L2(p) by its order and one special conjugacy class size

In the past thirty years, many authors investigated some quantitative characterizations of ﬁnite groups, especially ﬁnite simple groups, such as quantitative characterizations by group order and element orders, by the set of lengths of conjugacy classes, by dimensions of irreducible characters, etc. In this article the projective special linear group L 2 (p) is characterized by its order and one special conjugacy class size, where p ### Absent Data Generating Classifier for Imbalanced Class Sizes

We propose an algorithm for two-class classification problems when the training data are imbalanced. This means the number of training instances in one of the classes is so low that the conventional classification algorithms become ineffective in detecting the minority class. We present a modification of the kernel Fisher discriminant analysis such that the imbalanced nature of the problem is explicitly addressed in the new algorithm formulation. The new algorithm exploits the properties of the existing minority points to learn the effects of other minority data points, had they actually existed. The algorithm proceeds iteratively by employing the learned properties and conditional sampling in such a way that it generates sufficient artificial data points for the minority set, thus enhancing the detection probability of the minority class. Implementing the proposed method on a number of simulated and real data sets, we show that our proposed method performs competitively compared to a set of alternative state-of-the-art imbalanced classification algorithms. Keywords: kernel Fisher discriminant analysis, imbalanced data, two-class classification ### The exact number of conjugacy classes for 2 - generator p - groups of nilpotency class 2

The major contribution of this thesis will be the new theoretical results on the exact number of conjugacy classes for 2-generator p-groups of class 2 (p an odd prime). This thesis also contributes to a greater challenge of counting conjugacy classes of groups in general. No classes of nilpotent groups have formulas for their exact number of conjugacy classes. Therefore, this thesis provides original results. Some of the results have been presented in national and international conferences and thus contribute to new findings in the field of group theory. ### Class sizes in primary schools in Wales : September 2011

Last year the number of infant classes of over 30 pupils (and pupils within such classes) increased; there was a corresponding increase in the number of junior classes of over 30 pupils. The total number of infant and junior classes of over 30 pupils has risen a little over the past 7 years but remains lower than a decade ago when the statutory duties on local authorities in respect of infant class sizes were introduced. Chart 1: Percentage of Infant and Junior pupils in classes over 30 ### \mathversion{bold}ON NON-ABELIAN GROUPS OF ORDER \$2^n\$, \$n\geq 4\$ USING \textsf{GAP}

Let G be a finite group, and x ∈ G with o(x) = k. Then, we need to count all y ∈ G which has the same order as x. Precisely, we will determined a class of each order. The set of all of these classes is called the order classes of G, and denoted by OC G . This can be obtained by: The set of all ordered pairs [k, |O k |] ### Hecke Groups, Dessins d’Enfants and the Archimedean Solids

With the above in mind, let us begin our story by recalling that the Platonic solids are the convex polyhedra with equivalent faces composed of congruent convex regular polygons; these objects have been known and studied for millennia. These solids also appear in mathematical physics: for example, the symmetries of the Platonic solids are related to D-brane orbifold theories . Now, more broadly, another well-known class of convex polyhedra are the Archimedean solids: the semi-regular convex polyhedra composed of two or more types of regular polygons meeting in identical vertices, with no requirement that faces be equivalent. There are three categories of such Archimedean solids: (I) the Platonic solids; (II) two infinite series solutions – the prisms and anti- prisms; and (III) fourteen further exceptional cases. ### Primitive stability and Bowditch conditions for rank 2 free group representations

Remark 2.5. Observe that the two conditions above are invariant under conjugacy in P SL(2, C ), hence it makes sense to speak about classes in the character variety satisfying these requirements. The second BQ condition, in particular, implies that the absolute values of traces of primitive elements do not accumulate anywhere in R , that is we can replace the 2 in the inequality with any positive constant. This result has been proved by Bowditch and follows from Theorem 2 in [Bow]. This fact is highly non trivial, and the original argument makes use of the notion of Fibonacci growth for certain maps associated to Markoff triples. This machinery involves a number of topics and technicalities which go beyond the purpose of this paper. The interested reader can find more details in [Bow] and [TWZ], for instance. ### Asymptotics in conjugacy classes for free groups

In the same setting as his prime geodesic theorem, Huber  also established asymptotic results for certain groups of isometries of the hyperbolic plane. Here the connection to the prime geodesic theorem is as follows. Instead of counting prime closed geodesics on a surface, these results counted closed geodesics. We shall state Huber’s results in terms of the action of a group of isometries acting on the hyperbolic plane and so we use the term orbit counting in this setting. Huber also established an orbit counting result when the elements of the group are restricted to a non-trivial conjugacy class and this corresponds to counting based closed geodesics. We refer the reader to  for an introduction to the action of PSL(2, C) on the hyperbolic plane H 2 . A M¨ obius map f on the Riemann sphere has the form ### Classification of involutions of SL(n,k) and SO(2n+1,k)

For k = R the group SO(2n + 1, k) is k-anisotropic, so there are no (θ, k)-split tori, so consequently also no k-inner elements. By [Hel88, Lemma 10.3] the isomorphy classes of involutions of k-anisotropic real groups correspond to those of the corresponding complex group. Moreover for k algebraically closed or the real numbers there is only one isomorphy class of involutions I A with A = I s,t for each value of s. So also for this reason there are no k-inner elements in these two cases. In this section we will determine the k-inner elements for k = Q p , but first we list the admissible (Γ, σ)- indices for G = SO(2n + 1, k).  