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

11 Read more

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 [6]. 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.

Show more
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.

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.

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.

24 Read more

set X ∗ and two vertices m and n are connected if gcd(m, n) > 1. Motivated by [3], 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 [4] and the bipartite divisor graph for group **conjugacy** **class** **sizes** has been studied in [3] and [5]. The bipartite divisor graph B (X) is the graph with vertex set the disjoint union 1 ρ(X) ∪ X ∗ and edge set

The notion of the bipartite divisor graph was first introduced by Iranmanesh and Praeger in [4] for a finite set of positive integers. As an application of this graph in group theory, in [2], 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 [20] 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

Show more
11 Read more

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.

16 Read more

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.

Show more
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

10 Read more

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

Show more
30 Read more

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.

17 Read more

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

10 Read more

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 |]

12 Read more

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 [11]. 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.

Show more
29 Read more

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.

Show more
65 Read more

In the same setting as his prime geodesic theorem, Huber [25] 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 [1] 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

Show more
105 Read more

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).

Show more
92 Read more

the group of signed permutations; that is, permutations w of {1, . . . , n, −1, . . . , −n} such that w(−i) = −w(i) for 1 ≤ i ≤ n. Signed permutations can be written as permutations where each number has either a plus or a minus sign above it. So, for example, if w = ( + 1 − 2 − 3), then w(1) = 2, w(−1) = −2, w(2) = −3, w(−2) = 3, w(3) = −1 and w(−3) = 1. The set of signed permutations where an even number of minus signs appear is a subgroup which is of type D n . **Conjugacy** classes in types

17 Read more

Teachers say that teaching in smaller classes improve student-teacher relationship, strengthens the connection between a teacher and his/her student, making the experience much more enjoyable and meaningful to all participants. Having a small number of students in a given **class** makes it possible for teachers to become familiar with the students on a personal level. Precisely this feeling of familiarity helps students feel safe and beneficial in their education, which ultimately helps them to perform better in the classroom.