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Let Γ be a simple finite graph with vertex set V (Γ) and edge set E (Γ) . Let R be an equivalence relation on V (Γ) . The R -super Γ graph Γ R is a simple graph with vertex set V (Γ) and two distinct vertices are adjacent if either they are in the same R -equivalence class or there are elements in their respective R -equivalence classes that are adjacent in the original graph Γ . We first show that Γ R is a generalized join of some complete graphs and using this we obtain the adjacency and Laplacian spectrum of conjugacy super commuting graphs and order super commuting graphs of dihedral group D 2 n (n ≥ 3) , generalized quaternion group Q 4 m (m ≥ 2) and the nonabelian group Z p ⋊ Z q of order pq, where p and q are distinct primes with q | (p - 1) . [ABSTRACT FROM AUTHOR]

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]

Suppose G is a finite non-abelian group and (G) is a graph with non-central conjugacy classes of G as its vertex set. Two vertices L and K in such that ab = ba. This graph is called the commuting conjugacy class graph of G. The purpose of this paper is to compute the commuting conjugacy class graph of the finite 2 such that ab = ba. This graph is called the commuting conjugacy class graph of G. The purpose of this paper is to compute the commuting conjugacy class graph of the finite 2-group Gm(n) and G(n). [ABSTRACT FROM AUTHOR]

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

One of the important problem in finite groups theory is group characterization by specific property. Properties, such as element order, the set of element with the same order, etc. In this paper, we prove that Ree group ²G2(q), where q±√3q+1 is a prime number can be uniquely determined by its order and one conjugacy class size. [ABSTRACT FROM AUTHOR]

Let p be a prime and let G be a finite group such that the smallest prime that divides |G| is p. We find sharp bounds, depending on p, for the commuting probability and the average character degree to guarantee that G is nilpotent or supersolvable. [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]

In this paper, we determine all finite groups in which each element has order a power of a prime and have at most five non-central conjugacy classes with representatives of orders multiples of p, for each arbitrary prime number p. [ABSTRACT FROM AUTHOR]

Let G be a finite group. In this paper, we say that g ∈ G is an SM-vanishing element of G, if there exists a strongly monolithic character χ of G such that χ(g) = 0. The conjugacy class of an SM-vanishing element of G is called an SM-vanishing conjugacy class of G. Our purpose here is to prove that for determining some properties of the structure of the group G, it is enough to consider the same arithmetical conditions on the sizes of SM-vanishing conjugacy classes of G instead of certain arithmetical conditions on the sizes of vanishing conjugacy classes of G. [ABSTRACT FROM AUTHOR]

A constructive procedure is given to determine all ideals of a finite-dimensional solvable Lie algebra. This is used in determining all conjugacy classes of subalgebras of a given finite-dimensional solvable Lie algebra. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structure of finite solvable groups G admitting a normal subgroup N such that G / N is non-abelian and N contains all but four conjugacy classes of G. [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]

Let G be a finite AC-group such that [ G : Z (G) ] = 2 n s , where n ≥ 3 , s ≥ 1 and s is odd. We prove that if G has 2 n + 1 centralizers of elements, then s = 3 , n is an even integer, the set of the sizes of the conjugacy classes of G is { 1 , 2 n , 3 ⋅ 2 n − 1 } and G / Z (G) is a Frobenius group whose Frobenius kernel is an elementary abelian 2 -group of order 2 n and the Frobenius complement is a group of order 3. [ABSTRACT FROM AUTHOR]

In this paper, using a method of construction of $1$-designs which are not necessarily symmetric, introduced by Key and Moori, we determine a number of $1$-designs with interesting parameters from the maximal subgroups and the conjugacy classes of elements of the group $PSL(2,q)$ for $q$ a power of an odd prime.

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 F be the field with p elements, where p is of the form (2 t + 1) for some square free odd integer t. In this article, we obtain the order of the symmetric and the unitary subgroup of U (F C q) , where q is a prime divisor of t. Consequently, we resolve the normal complement problem for the modular group algebra of a split extension of C q by an abelian group of order p m with m ≥ (q - 3) , over the field with p elements such that p = (2 q + 1). Further, we study the normal complement problem in the finite semisimple group algebras of general linear groups. [ABSTRACT FROM AUTHOR]

Let $ A $ be a finite-dimensional algebra with identity over the field $ \mathbb{F} $, $ U(A) $ be the group of units of $ A $ and $ L(A) $ be the set of left ideals of $ A $. It is well known that there is an equivalence relation $ \sim $ on $ L(A) $ by defining $ L_1\sim L_2\in L(A) $ if and only if there exists some $ u\in U(A) $ such that $ L_{1} = L_{2}u $. $ C(A) = \{[L]|L\in L(A)\} $ is the set of equivalence classes determined by the relation $ \sim $, which is a semigroup with respect to the natural operation $ [L_1][L_2] = [L_1L_2] $ for any $ L_1, L_2 \in L(A) $. The purpose of this paper is to describe the structures of semigroup of conjugacy classes of left ideals for the Sweedler's four-dimensional Hopf algebra $ H_{4} $.

The extended conjugate graph associated to a finite group $ G $ is defined as an undirected graph with vertex set $ G $ such that two distinct vertices joined by an edge if they are conjugate. In this article, we show that several properties of finite groups can be expressed in terms of properties of their extended conjugate graphs. In particular, we show that there is a strong connection between a graph-theoretic property, namely regularity, and an algebraic property, namely nilpotency. We then give some sufficient conditions and necessary conditions for the non-central part of an extended conjugate graph to be regular. Finally, we study extended conjugate graphs associated to groups of order $ pq $, $ p^3 $, and $ p^4 $, where $ p $ and $ q $ are distinct primes.

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]

In this paper we characterize, in terms of their conjugacy classes, linear groups G such that G / ζ k (G) belongs to a certain group class X for several natural choices of X . Moreover, a description is given of linear groups with restrictions on layers. [ABSTRACT FROM AUTHOR]

Given a finite group G and x ∈ G, the class size of x in G is called odd-square-free if it is not divisible by the square of any odd prime number. In this paper, we show that if G is a nonsolvable finite group, all of whose class sizes are odd-square-free, then we have some control on the structure of G, which is an answer to the dual of the question mentioned by Huppert in [5]. [ABSTRACT FROM AUTHOR]

A graph is called the divisibility graph of if its vertex set is the set of noncentral conjugacy class sizes of and there is an edge between vertices and if and only if or . We determine the number of connected components of the divisibility graph when is an F-group. A finite group is called an F-group if for every , implies . We also prove that if the divisibility graph in which is an F-group is a -regular graph, then the divisibility graph is a complete graph with vertices. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the structure and sizes of conjugacy classes in certain affine semi-linear groups. This provides a wealth of finite groups of small conjugate rank that are solvable and non-nilpotent.

The aim of this paper is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of commutativity degree of a group algebra F[G], where G is a finite group and F is a finite field. We prove that two isoclinic groups for which the order of their centers are equal have the same commutativity degree. Finally, we give some lower and upper bounds for the commutativity degree of group algebra F[G] in terms of the order of G, the order of F and d(G). [ABSTRACT FROM AUTHOR]

Let G be a non-abelian finite simple group. A famous result of Liebeck and Shalev is that there is an absolute constant c such that whenever S is a non-trivial normal subset in G then S k = G for any integer k at least c · (log | G | / log | S |) . This result is generalized by showing that there exists an absolute constant c such that whenever S 1 , ... , S k are normal subsets in G with ∏ i = 1 k | S i | ≥ | G | c then S 1 ⋯ S k = G . [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called commensurable with a normal subgroup (cn) if there is N C G such that |HN/(H ∩ N)| is finite. We characterize generalized radical groups G which have one of the following finiteness conditions: (A) the minimal condition on non-cn subgroups of G; (B) the non-cn subgroups of G fall into finitely many conjugacy classes; (C) the non-cn subgroups of G have finite rank.

Let G be the alternating group Alt (n) on n letters. We prove that for any ε > 0 , there exists N = N (ε) ∈ N such that whenever n ≥ N and A, B, C, D are normal subsets of G each of size at least | G | 1 / 2 + ε , then A B C D = G . [ABSTRACT FROM AUTHOR]

In this paper, we study the Q-conjugacy character table of an arbitrary finite group and introduce a general relation between the degrees of Q-conjugacy characters with their corresponding reductions. This could be accomplished by using the Hermitian symmetric form. We provide a useful technique to calculate the character table of a finite group when its corresponding Qconjugacy character table is given. Then, we evaluate our results in some useful examples. Finally, by using GAP (Groups, Algorithms and Programming) package, we calculate all the dominant classes of the sporadic Conway group Co2 enabling us to find all possible the integer-valued characters for the Conway group Co2. [ABSTRACT FROM AUTHOR]

In this paper we study the locus of generalized degree d Hénon maps in the parameter space RatdN of degree d rational maps PN →PN modulo the conjugation action of SLN+1. We show that Hénon maps are never in the GIT stable locus, that they are in the GIT unstable locus if d ≥ 3, and that they are in the strictly GIT semistable locus if N = d = 2. We also give a general classification of all unstable maps in Rat22. [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]

We show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSLn(q) and PSp2n(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one class of involutions in PSUn(2m). [ABSTRACT FROM AUTHOR]

Let G = H × A be a finite 2-group, where H is a non-abelian group of order 8 and A is an elementary abelian 2-group. We obtain a normal complement of G in the normalized unit group V(FG) and in the unitary subgroup V ∗ (F G) over the field F with 2 elements. Further, for a finite field F of characteristic 2, we derive class size of elements of V(FG). Moreover, we provide class representatives of V ∗ (F H) . [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A -modules have dimension at least 6. [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]

Let G be a finite group and δ (G) denote the number of conjugate classes of allnon-cyclic subgroups of G. The symbol π (G) denotes the set of the prime divisors of |G|. In Meng and Li (Sci Sin Math 44:939–944, 2014), it was proved that for a finite non-cyclic solvable group G, one always has δ (G) ≥ 2 | π (G) | - 2 . The groups with δ (G) ≤ | π (G) | + 1 always are solvable and have been complete classified. Moreover, it was showed that a finite non-solvable group G with δ (G) = | π (G) | + 2 is isomorphic to A 5 or SL(2, 5). In this paper, we investigate the finite solvable groups with δ (G) = | π (G) | + 2 . For convenience, a group G is said to be a δ π 2 -group if δ (G) = | π (G) | + 2 . In particular, we give a completely classification of the δ π 2 -groups with | π (G) | = 3 , 4 . [ABSTRACT FROM AUTHOR]

In this work, a conjugacy class graph is represented by Γ and G represents non-abelian metabelian group. Conjugacy class graph of a group is the graph associated with the conjugacy classes of the group. Its vertices are the non-central conjugacy classes of the group, and two distinct vertices are joined by an edge if their cardinalities are not coprime. A group is referred to as metabelian if there exits an abelian normal subgroup in which the factor group is also abelian. It has been proven earlier that 25 non-abelian metabelian groups which have order less than 24, which are considered in this work, exist. In this article, the conjugacy class graphs of non-abelian metabelian groups of order less than 24 are determined as well as examples of some finite groups associated to other graphs are given. [ABSTRACT FROM AUTHOR]

Let F be a finite field of characteristic p > 0. In this article, we obtain a relation between the class length of elements of a finite p-group G in the normalized unit group V(FG) and its unitary subgroup V*(FG), when p is an odd prime. We also provide the size of the conjugacy class of non-central elements of a group G in V(FG), where either G is any finite p-group with nilpotency class 2 or G is a p-group with nilpotency class 3 such that |G| ≤ p5. [ABSTRACT FROM AUTHOR]

Recently, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy)=o(x)o(y) for every x,y∈G of coprime order. Motivated by this result, we study the groups with the property that (xy)G=xGyG and those with the property that χ(xy)=χ(x)χ(y) for every χ∈Irr(G) and every nontrivial x,y∈G of pairwise coprime order. We also consider several ways of weakening the hypothesis on x and y. While the result of Baumslag and Wiegold is completely elementary, some of our arguments here depend on (parts of) the classification of finite simple groups. [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]

Let G be the special linear group SL(2; q). We show that if (a1,..., at) and (b1,..., bt) are sampled uniformly from large subsets A and B of Gt, then their interleaved product a1b1a2b2 ... atbt is nearly uniform over G. This extends a result of the first author [W. T. Gowers, Combin. Probab. Comput., 17 (2008), pp. 363-387], which corresponds to the independent case where A and B are product sets. We obtain a number of other results. For example, we show that if X is a probability distribution on Gm such that any two coordinates are uniform in G², then a pointwise product of s independent copies of X is nearly uniform in Gm, where s depends on m only. Extensions to other groups are also discussed. We obtain closely related results in communication complexity, which is the setting where some of these questions were first asked by Miles and Viola [Shielding circuits with groups, in ACM Symposium on the Theory of Computing (STOC), ACM, New York, 2013, pp. 251-260]. For example, suppose party Ai of k parties A1,...,Ak receives on its forehead a t-tuple (ai1,..., ait) of elements from G. The parties are promised that the interleaved product a11...ak1a12...ak2...a1t...akt is equal either to the identity e or to some other fixed element g 2 G, and their goal is to determine which of the two the product is equal to. We show that for all fixed k and all sufficiently large t the communication is (t log jGj), which is tight. Even for k = 2 the previous best lower bound was (t). As an application, we establish the security of the leakage-resilient circuits studied by Miles and Viola [Shielding circuits with groups, in ACM Symposium on the Theory of Computing (STOC), ACM, New York, 2013, pp. 251{260] in the "only computation leaks" model. [ABSTRACT FROM AUTHOR]

In this paper, we establish statistical results for a convex co-compact action of a free group on a CAT( - 1 ) space where we restrict to a non-trivial conjugacy class in the group. In particular, we obtain a central limit theorem where the variance is twice the variance that appears when we do not make this restriction. [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]

We use elementary finite group theory to give direct and short proofs related to the results of conjugacy classes of finite groups. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation.