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Given a finite group G, denote by cs(G) the set of the sizes of the conjugacy classes of G and by Cent(G) the set of the centralizers of elements of G. Consider a prime p and integers s ≥ 2 and n ≥ 2 , with gcd (p , s) = 1 . In this paper, some relations between cs(G) and | Cent (G) | are established in the case where cs (G) = { 1 , p n , p n − 1 s } . Further, when p ∈ { 2 , 3 } , we determine the values of s and the structure of a finite group G such that cs (G) = { 1 , p n , p n − 1 s } . We also describe the structure of an AC-group G such that [ G : Z (G) ] = 3 n s and | Cent (G) | = 1 + ∑ i = 0 n 3 i , where gcd (3 , s) = 1 . [ABSTRACT FROM AUTHOR]

There is a well-known classification of conjugacy classes of involutions in finite Coxeter groups, in terms of subsets of nodes of their Coxeter graphs. In many cases, the product of an involution with the longest element w o is again an involution. We identify the conjugacy class of this product involution in terms of said classification. Communicated by Mandi Schaeffer Fry [ABSTRACT FROM AUTHOR]

Let m be a positive integer such that p ∤ m where p is prime. In this paper we find the number of conjugacy classes of completely reducible cyclic subgroups in GL (2 , q) of order m, where q is a power of p. [ABSTRACT FROM AUTHOR]

For a simple algebraic group G over an algebraically closed field we study products of normal subsets. For this we mark the nodes of the Dynkin diagram of G. We use two types of labels, a binary marking and a labeling with nonnegative integers. The first is used to recognize large conjugacy classes which appear in a product of two conjugacy classes while the second is used to keep track of multiplicities of regular diagrams. In particular, we formulate sufficient conditions in terms of marked diagrams, for a product of normal subsets in G to contain regular semisimple elements. [ABSTRACT FROM AUTHOR]

Let p and q be odd primes such that p = 2 q + 1. Let F be the field with p elements and G = A ⋊ C q be a group, where A is an abelian group of order p m. In this article, we prove that if m ≥ (q − 1) , then G does not have a normal complement in U (F G). Further, for any integer n ≥ 4 , we prove that if F is a finite field such that char (F) > n , then S n and A n do not have a normal complement in U (F S n) and U (F A n) , respectively. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and G be a finite split metabelian p-group of exponent p. In this article, we obtain a normal complement of G in V (F G) , where F is the field with p elements. Further, assume that G = A ⋊ C 3 , where A is a finite abelian p-group and 3 | p − 1. If F is any finite field of characteristic p, then we prove that G does not have a normal complement in V (F G) and obtain the structure of the unitary subgroup V * (F G). Communicated by Sudarshan Kumar Sehgal [ABSTRACT FROM AUTHOR]

For a finite group G, we define an equivalence relation ∼ on G as follows: where xG is the conjugacy class of x in G. The set of sizes of equivalence classes with respect to this relation is called the same-size conjugate set of G and denote it by U(G). In this paper, we consider the influence of U(G) on the structure of a group G. In fact, we conjecture that all finite non-abelian simple groups G are characterizable by U(G) and we confirm this conjecture for the projective special linear groups PSL3(3) and PSL2(q), where. [ABSTRACT FROM AUTHOR]

In this article, we discuss the normal complement problem for metacyclic groups in modular group algebras. If F is the field with p elements and G is a finite split metacyclic p-group of nilpotency class 2, then we prove that G has a normal complement in For a finite field F of characteristic p, where p is an odd prime, we prove that has a normal complement in if and only if p = 3 and [ABSTRACT FROM AUTHOR]

A construction of primitive 1-designs invariant under the action of a finite primitive group is used to determine the parameters of certain symmetric 1-designs. Using a method referred to as Key–Moori Method 1, in this article we construct a number of symmetric 1-designs from the maximal subgroups of the small Ree groups. [ABSTRACT FROM AUTHOR]

Given a finite group G, denote by N(G) the set of sizes of its conjugacy classes. We show that if G has trivial center and N(G) equals or for then G has a composition factor isomorphic to the alternating group Altk for some such that the half-interval contains no primes. As a corollary, we prove Thompson's conjecture is valid for simple alternating groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group and δ(G) denote the number of conjugacy classes of all non-cyclic subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In [7], Meng and Li showed the inequality δ(G)≥2|π(G)|−2, where G is non-cyclic solvable group. In this paper, we describe the finite groups G such that δ(G) = 2|π(G)|−2. Another aim of this paper would show δ(G)≥M(G)+2 for unsolvable groups G and the equality holds ⇔G≅A5 or SL(2,5), where M(G) denotes the number of conjugacy classes of all maximal subgroups of G. [ABSTRACT FROM AUTHOR]

In this paper, we use Key-Moori methods 1 and 2 to construct some designs from the maximal subgroups and conjugacy classes of the groupPSL2(q), whereqis a power of 2. [ABSTRACT FROM PUBLISHER]

Aweak Cayley table isomorphismis a bijectionφ:G→Hof groups such thatφ(xy)∼φ(x)φ(y) for allx,y∈G. Here ∼ denotes conjugacy. WhenG = Hthe set of all weak Cayley table isomorphismsφ:G→Gforms a group𝒲(G) that contains the automorphism groupAut(G) and the inverse mapI:G→G,x↦x−1. Let𝒲0(G) = ⟨Aut(G),I⟩≤𝒲(G) and say thatGhastrivial weak Cayley table groupif𝒲(G) = 𝒲0(G). We show thatPSL(2,pn) has trivial weak Cayley table group, wherep≥5 is a prime andn≥1. [ABSTRACT FROM PUBLISHER]

A weak Cayley table isomorphism is a bijection φ : G → H of groups such that φ(xy) ~ φ(x)φ(y) for alt x,y e G. Here ~ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ : G → G forms a group W(G) that contains the j automorphism group Aut(G) and the inverse map I : G → G,x t-> x-1. Let W0(G) = (Aut(G), I) < W(G) and say that G has triviaI weak Cayley table group if W(G) = W0(G). We show that all finite irreducible Coxeter groups (except possibly E8) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups. [ABSTRACT FROM AUTHOR]

LetGbe a finite group and τ(G) denote the number of conjugacy classes of all non-abelian subgroups ofG. The symbol π(G) denotes the set of the prime divisors of |G|. In this paper, finite groups with τ(G) ≤ |π(G)| are classified completely. Furthermore, finite nonsolvable groups with τ(G) = |π(G)| +1 are determined. [ABSTRACT FROM AUTHOR]

The work is inspired by an article of Herzog, Longobardi, and Maj, who considered groups with a finite number of infinite conjugacy classes. Their main results were obtained under assumption that theFC-center is of finite index in the group. We consider here infinite groups with a finite number of conjugacy classes of any size (FNCC-groups). Hence theFC-center in our case will be finite, but of infinite index in the group. Among results on these groups we give a criterion for a wreath product ofFNCC-groups to be anFNCC-group. [ABSTRACT FROM AUTHOR]

LetGbe a finite group andM(G) be the subgroup ofGgenerated by all noncentral elements ofGthat lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class ofM(G) andF(M(G)), whereF(M(G)) denotes the Fitting subgroup ofM(G). We prove some conditional results regarding the nilpotency class ofM(G). [ABSTRACT FROM PUBLISHER]

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail examples involving the ring of integers modulo n. If R is a local ring with maximal ideal m such that R/m = ℤ2 and m 2 = (0), we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G. [ABSTRACT FROM PUBLISHER]

As an important application of Thompson's theorem [9, Theorem 10.4.2], a finite group is solvable if it has an abelian maximal subgroup. In this article, we mainly investigate the influence of some quantitative properties of abelian subgroups on solvability of finite groups. Some new results are obtained. [ABSTRACT FROM PUBLISHER]

Let K be a conjugacy class of a finite p-group G where p is a prime, and let K-1 denote the conjugacy class of G consisting of the inverses of the elements in K. We observe that, in several cases, the number of elements in the product KK-1 is congruent to 1 modulo p - 1, and we pose the question in which generality this congruence is valid. We also consider related properties of the class multiplication constants of G. Furthermore, let χ be an irreducible character of G, and let [image omitted] denote the complex conjugate of χ. We show that, in several cases, the number of irreducible constituents of the product [image omitted] is congruent to 1 modulo p - 1, and we pose the question in which generality this congruence is valid. [ABSTRACT FROM AUTHOR]

In this article, we show that for every abelian subgroup H of a Garside group, some conjugate g-1Hg consists of ultra summit elements and the centralizer of H is a finite index subgroup of the normalizer of H. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning abelian subgroups of Garside groups. [ABSTRACT FROM AUTHOR]

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture. Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G):={n ∈  | G has a conjugacy class of size n}. In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows. AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ∇(G) ≅ ∇ (M). Then G ≅ M. In this short article we prove that if G is a finite group with ∇(G) ≅ ∇ (A10), then G ≅ A10, where A10 is the alternating group of degree 10. [ABSTRACT FROM AUTHOR]

The aim of this article is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of relative commutativity degree of a subgroup H of a group G (denoted by d(H, G)). We shall state some results concerning the new concept which are mostly new or improvements of known results given in Gustafson (1973) and Moghaddam et al. (2005). Moreover, we shall define the relative nth nilpotency degree of a subgroup of a group and give some results concerning this at the end of the article. [ABSTRACT FROM AUTHOR]

The aim of this note is to classify the finite solvable groups with an irreducible character that vanishes on just one class of elements. [ABSTRACT FROM AUTHOR]

Let k(G) be the number of conjugacy classes of finite groups G and πe(G) be the set of the orders of elements in G. Then there exists a non-negative integer k such that k(G) = |πe(G)| + k. We call such groups to be co(k) groups. This article classifies all finite co(1) groups. They are isomorphic to one of the following groups: A5, L2(7), S5, Z3, Z4, S5, A4, D10, Hol(Z5), or Z3 x Z4. [ABSTRACT FROM AUTHOR]

Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If [Xfr] is a class of groups, a group G is said to have [Xfr]-conjugate classes of subgroups if G/CoreG(NG(H)) ∈ [Xfr] for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroup if and only if it is central-by-(polycyclic-by-finite). [ABSTRACT FROM AUTHOR]