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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]

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]

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]

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]

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 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]

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]

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]

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]

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]

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]

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 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]

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]

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]

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]

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]

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]

A group class X is said to be countably recognizable if a group belongs to X whenever all its countable subgroups lie in X. It is proved here that most of the relevant classes of groups defined by restrictions on the conjugacy classes are countably recognizable. [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]

Giving a condition for the amenability of groups, Rosenblatt and Willis first introduced the concept of configuration. From the beginning of the theory, the question whether the concept of configuration equivalence coincides with the concept of group isomorphism was posed. We negatively answer this question by introducing two non-isomorphic, solvable and hence amenable groups which are configuration equivalent. Also, we will prove this conjecture, due to Rosenblatt and Willis, whether the configuration equivalent groups include the free non-Abelian group of the same rank or not. We show that two-sided equivalent groups have same class numbers. [ABSTRACT FROM AUTHOR]

Let $w$ be a group-word. For a group $G$, let $G_{w}$ denote the set of all $w$-values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$. It is known that if $w$ is a concise word, then $G$ is an $FC(w)$-group if and only if $w(G)$ is $FC$-embedded in $G$, that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$. There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$, we show that if $G$ is an $FC(w)$-group, then the commutator subgroup $w(G)^{\prime }$ is $FC$-embedded in $G$. We also establish the analogous result for $BFC(w)$-groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite. [ABSTRACT FROM AUTHOR]

We study the semigroup structure on the set of conjugacy classes of left ideals of a finite-dimensional algebra over an algebraically closed field , equipped with the natural multiplication inherited from , and the structure of the contracted semigroup algebra . It is shown that has a finite chain of ideals with either nilpotent or completely -simple factors with trivial maximal subgroups, so in particular it is locally finite. The ordinary quiver of is proved to be a subquiver of , if is finite. Moreover, in this case, the structure of determines, up to isomorphism, the structure of the algebra modulo its Jacobson radical. Combining these results we show that if the semigroup is finite, then it determines the structure of any (not necessarily basic) triangular algebra which admits a normed presentation. [ABSTRACT FROM AUTHOR]

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]

The nilpotent conjugacy class graph (or NCC-graph) of a group G is a graph whose vertices are the nontrivial conjugacy classes of G such that two distinct vertices xG and yG are adjacent if ⟨x',y'⟩ is nilpotent for some x' Є xG and y' Є yG. We discuss on the number of connected components as well as diameter of connected components of these graphs. Also, we consider the induced subgraph Γn (G) of the NCC-graph with vertices set {gG \ g Є G \ Nil (G)}, where Nil(G) = {g Є G \ ⟨x, g⟩ is nilpotent for all x Є G}, and classify all nite non-nilpotent group G with empty and triangle-free NCC-graphs. [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]

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a conjugacy class collapses and prove that for infinitely many pairs (n, q), any finite-dimensional pointed Hopf algebra H with G(H) ≃ PSLn(q) or SLn(q) is isomorphic to a group algebra. [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]

In this paper, we classify the conjugacy classes of the action of PSL₂(O₂) on the projective line over finite fields, PL(F_{p}), where p is the M-S prime, by using the method of parametrization and investigate the behavoior of coset diagrams of these actions. We prove that the action of PSL₂(O₂) on PL(F_{p}) is transitive for all conjugacy classes except for the conjugacy class in which 2 is a perfact square in F_{p}. We also prove that the homomorphic images of PSL₂(O₂) represented by these coset diagrams are isomorphic to the rank one Chevalley groups, L₂(p), for all p≥11. We also study the behavior of the coset diagram of the homomorphic images of PSL₂(O₂) for the conjugacy class in which 2 is a perfact square in F_{p} and prove that these coset diagrams admit symmetry about the vertical line of axis in two dimensional space. We also prove that these coset diagrams depict intransitive action of PSL₂(O₂) on PL(F_{p}). This algebraic fact leads us to develop a formula to count the number of orbits occuring in each coset diagram of this particular class. [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)|

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups. [ABSTRACT FROM AUTHOR]

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]

Two basic results on S-rings over an Abelian group are the Schur theorem on multipliers and the Wielandt theorem on primitive S-rings over groups with a cyclic Sylow subgroup. Neither of these is directly generalized to the non-Abelian case. Nevertheless, we prove that the two theorems are true for central S-rings over any group, i.e., for S-rings that are contained in the center of the group ring of that group (such S-rings arise naturally in the supercharacter theory). Extending the concept of a B-group introduced by Wielandt, we show that every Camina group is a generalized B-group, whereas simple groups, with few exceptions, cannot be of this type. [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]

The density of each conjugacy class in the space of mixing Z-actions is proved. This result implies the genericity of rank 1, the triviality of the centralizer, and the absence of factors. [ABSTRACT FROM AUTHOR]