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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]

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]

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]

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]

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]

We generalize two results about conjugacy class sizes of real elements to sub-class sizes of real elements. [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]

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]

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]

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]

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]

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]

In this paper we introduce the graph Γ s c (G) associated with a group G , called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of G and two distinct conjugacy classes C , D are adjacent if there exist x ∈ C and y ∈ D such that 〈 x , y 〉 is solvable. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number d , and we find explicitly the list of such groups with d = 2. We pose some problems on the relation of the SCC-graph to the solvable graph and to the NCC-graph, which we cannot solve. [ABSTRACT FROM AUTHOR]

Given a finite group G, we denote by $$\psi (G)$$ the sum of the element orders in G. In this article, we prove that if t is the number of nonidentity conjugacy classes in G, then $$\psi (G)=1+t|G|$$ if and only if G is either a group of prime order or a nonabelian group of the square-free order with two prime divisors. Also we find a unique group with the second maximum sum of the element orders among all finite groups of the same square-free order. [ABSTRACT FROM AUTHOR]

In this paper we prove that a finite group G having at most three conjugacy classes of non-normal non-abelian proper subgroups is always solvable except for G ≅ A5, which extends Theorem 3.3 in [Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable, Acta Math. Sinica (English Series) 27 (2011) 891-896.]. Moreover, we show that a finite group G with at most three same order classes of non-normal non-abelian proper subgroups is always solvable except for G ≅ A5. [ABSTRACT FROM AUTHOR]

A group G is said to be a V -group if every subnormal subgroup of G has only finitely many conjugates. It is proved here that if G is a group admitting an ascending normal series whose factors have finite rank, and all proper subgroups of G have the V -property, then G itself is a V -group, provided that G belongs to a suitable class of generalized soluble groups, containing in particular all locally (soluble-by-finite) groups. On the other hand, an example shows that there exist periodic metabelian minimal non-V groups. [ABSTRACT FROM AUTHOR]

In this paper, we consider all the Janko sporadic groups J1, J2, J3 and J4 (with orders 175560, 604800, 50232960 and 86775571046077562880, respectively) with a new concept called the markaracter- and Q-conjugacy character tables, which enables us to discuss marks and characters for a finite group on a common basis of Q-conjugacy relationships between their cyclic subgroups. Then by using GAP (Groups, Algorithms and Programming) package we calculate all their dominant classes enabling us to find all possible Q-conjugacy characters for these sporadic groups. Finally, we prove in a main theorem that all twenty six simple sporadic groups are unmatured. [ABSTRACT FROM AUTHOR]

‎Let G be a finite group‎. ‎An element g∈G is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character χ of G ‎, ‎χ(g)≠0‎. ‎The bi-Cayley graph BCay(G,T) of G with respect to a subset T⊆G‎, ‎is an undirected graph with‎ ‎vertex set G×{1,2} and edge set {{(x,1),(tx,2)}∣x∈G‎,‎ t∈T}‎. ‎Let nv(G) be the set‎ ‎of all non-vanishing elements of a finite group G ‎. ‎We show that g ∈ nv(G) if and only if the adjacency matrix of BCay(G,T)‎, ‎where T=Cl(g) is the‎ ‎conjugacy class of G ‎, ‎is non-singular‎. ‎We prove that ‎if the commutator subgroup of G has prime order p‎, ‎then‎ ‎(1) g ∈ nv(G) if and only if|Cl(g)|

Let G be a finite group and let K be the conjugacy class of x ∈ G. If K² is a conjugacy class of G, then [x, G] is solvable. If the order of x is a power of prime, then [x, G] has a normal p-complement. We also prove some related results on the solvability of certain normal subgroups when a non-trivial coset has certain properties. [ABSTRACT FROM AUTHOR]

Every finite group whose order is divisible by a prime p has at least 2 p − 1 conjugacy classes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

It is proved here that if G is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then G is either locally supersoluble or a Cernikov group. The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. As a consequence, it is shown that any in nite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble [ABSTRACT FROM AUTHOR]

For a finite group G, let γ(G) denote the number of conjugacy classes of all non-nilpotent subgroups of G, and let π(G) denote the set of the prime divisors of |G|. In this paper, we establish lower bounds on γ(G). In fact, we show that if G is a finite solvable group, then γ(G) = 0 or γ(G) ≥ 2|π(G)|-2, and if G is non-solvable, then γ(G) ≥ |π(G)| + 1. Both lower bounds are best possible. [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]

Let Fpn G be the group algebra of a finite p-group G over a field FPn with pn elements, and V(FPnG) the subgroup of units of augmentation 1. We investigate the conjugacy classes of V(Pn G), showing that p2n divides the order of the conjugacy class ∣ Ca ∣ for a noncentral element a e Fpn G and V(Fpn G) contains a conjugacy class of order p 2n, if a nonabelian finite p-group G has a factor group G/H such that its centre is of index p2. [ABSTRACT FROM AUTHOR]

Let G be a finite group and v(G) denote the number of conjugacy classes of non-normal subgroups of G. In this paper, all nilpotent groups G with v(G) = 3 are classified. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let x denote the conjugacy class of an element x of G. We classify all finite groups G in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G. [ABSTRACT FROM AUTHOR]

Finite nilpotent groups having exactly four conjugacy classes of non-normal subgroups are classified. [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]

Abstract: We introduce the notion of an ‘inverse property’ (IP) quandle which we propose as the right notion of ‘Lie algebra’ in the category of sets. For any IP-quandle we construct an associated group . For a class of IP-quandles which we call ‘locally skew’, and when is finite, we show that the noncommutative de Rham cohomology is trivial aside from a single generator θ that has no classical analogue. If we start with a group G then any subset which is ad-stable and inversion-stable naturally has the structure of an IP-quandle. If also generates G then we show that with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that this ‘covering map’ is an isomorphism for all finite crystallographic reflection groups W with the set of reflections, and that is locally skew precisely in the simply laced case. This implies that when W is simply laced, proving in particular a conjecture for in Majid (2004) [12]. We also consider as a locally skew IP-quandle ‘Lie algebra’ of and show that , the braid group on 3 strands. The map which therefore arises naturally as a covering map in our theory, coincides with the restriction of the usual universal covering map to the inverse image of . [Copyright &y& Elsevier]

It is shown that if G is a group of even order with trivial center such that ΙGΙ > 2ΙCG(t)Ι3 for some involution t ∈ G, then there exists a proper subgroup H of G such that ΙGΙ < ΙHΙ2. If ΙGΙ > ΙCG(t)Ι3 and k(G) is the class number of G, then ΙGΙ ≤ k(G)3/Ι [ABSTRACT FROM AUTHOR]

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]

Abstract: It has been proved recently by Moretó (2007) and Craven (2008) that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups by Jaikin-Zapirain (2005) . In this note, we prove that if G is a finite simple group then the order of G, denoted by , is bounded in terms of the largest multiplicity of its conjugacy class sizes. As a consequence, we prove that if the largest multiplicity of conjugacy class sizes of any quotient of a finite group G is m, then is bounded in terms of m. [Copyright &y& Elsevier]