Non-central Conjugacy Classes Research Articles

Conjugacy Class Graph of Some Non-Abelian Groups

The conjugacy class graph of a group G is a graph whose vertices are the non-central conjugacy classes of G and two vertices are adjacent if their cardinalities are not co-prime. In this paper, conjugacy class graphs of Dn, Q4n, Sn are studied. These graphs are found to be either complete graphs or union of complete graphs. Conjugacy classes of Dn × Dm are calculated and the results obtained are used to determine the structure of conjugacy class graphs of Dn × Dm, for odd and even values of m and n. Conjugacy class graphs of Dn are non-planar for n = 8 and n ≥ 11. They are non-hyperenergetic for all n and hypoenergetic only for n = 3, 5. Also, line graphs of these graphs are regular and eulerian for n ≡ 1 (mod 2) and n ≡ 0 (mod 4). The conjugacy class graphs of Q4n are non-planar for n = 4 and n ≥ 6. These graphs are non-hyperenergetic as well as non-hypoenergetic. The line graphs are eulerian for even values of n. It is conjectured that conjugacy class graph of Sn is non-planar for n ≥ 5.

Finite Groups with at Most Nine Non-Central Conjugacy Classes

Abstract: We classify all finite groups with at most nine non-central conjugacy classes.

The commuting conjugacy class graphs of finite groups with a given property

Abstract Let G be a finite non-abelian group. The commuting conjugacy class graph Γ ⁢ ( G ) {\Gamma(G)} is defined as a graph whose vertices are non-central conjugacy classes of G and two distinct vertices X and Y in Γ ⁢ ( G ) {\Gamma(G)} are connected by an edge if there exist elements x ∈ X {x\in X} and y ∈ Y {y\in Y} such that x ⁢ y = y ⁢ x {xy=yx} . In this paper, the structure of the commuting conjugacy class graph of group G with the property that G Z ⁢ ( G ) {\frac{G}{Z(G)}} is isomorphic to a Frobenius group of order pq or p 2 ⁢ q {p^{2}q} , is determined.

On conjugacy classes in groups

Let G be a group. Write G⁎=G﹨{1}.An element x of G⁎ will be called deficient if 〈x〉<CG(x) and it will be called non-deficient if 〈x〉=CG(x).If x∈G is deficient (non-deficient), then the conjugacy class xG of x in G will be also called deficient (non-deficient).Let j be a non-negative integer. We shall say that the group G has defect j, denoted by G∈D(j) or by the phrase “G is a D(j)-group”, if exactly j non-trivial conjugacy classes of G are deficient.

On the Topological Generation of Exceptional Groups by Unipotent Elements

AbstractLet G be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $$p \geqslant 0$$ p ⩾ 0 which is not algebraic over a finite field. Let $$\mathcal {C}_1, \ldots , \mathcal {C}_t$$ C 1 , … , C t be non-central conjugacy classes in G. In earlier work with Gerhardt and Guralnick, we proved that if $$t \geqslant 5$$ t ⩾ 5 (or $$t \geqslant 4$$ t ⩾ 4 if $$G = G_2$$ G = G 2 ), then there exist elements $$x_i \in \mathcal {C}_i$$ x i ∈ C i such that $$\langle x_1, \ldots , x_t \rangle $$ ⟨ x 1 , … , x t ⟩ is Zariski dense in G. Moreover, this bound on t is best possible. Here we establish a more refined version of this result in the special case where $$p&gt;0$$ p &gt; 0 and the $$\mathcal {C}_i$$ C i are unipotent classes containing elements of order p. Indeed, in this setting we completely determine the classes $$\mathcal {C}_1, \ldots , \mathcal {C}_t$$ C 1 , … , C t for $$t \geqslant 2$$ t ⩾ 2 such that $$\langle x_1, \ldots , x_t \rangle $$ ⟨ x 1 , … , x t ⟩ is Zariski dense for some $$x_i \in \mathcal {C}_i$$ x i ∈ C i .

Solvable conjugacy class graph of groups

In this paper we introduce the graph Γsc(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.

Covering Numbers for Simple Algebraic Groups

Let G be a simple algebraic group over an algebraically closed field, and let C be a noncentral conjugacy class of G. The covering number cn(G,C) is defined to be the minimal k such that G = Ck, where Ck = {c1c2⋯ck : ci ∈ C}. We prove that cn(G,C) le c frac {dim G}{dim C}, where c is an explicit constant (at most 120). Some consequences on the width and generation of simple algebraic groups are given.

Conjugacy classes of small size in solvable groups

If G is a finite group and , we say that x lies in a small class if is minimal among the sizes of the noncentral conjugacy classes of G. It has been conjectured that if G is a solvable group with trivial center and x belongs to a small class, then x lies in the center of the Fitting subgroup of G. We restrict the structure of a possible counterexample to this conjecture. We discuss the possible existence of a counterexample. As a consequence, we prove the conjecture when the small classes have prime sizes and also when all chief factors of G have rank at most 2. Perhaps surprisingly, the proof in the case when the small classes have prime size and the discussion on the existence of counterexamples use techniques from linear algebra.

An Arad and Fisman’s Theorem on Products of Conjugacy Classes Revisited

A theorem of Z. Arad and E. Fisman establishes that if A and B are two non-trivial conjugacy classes of a finite group G such that either AB=Acup B or AB=A^{-1} cup B, then G cannot be a non-abelian simple group. We demonstrate that, in fact, langle Arangle = langle Brangle is solvable, the elements of A and B are p-elements for some prime p, and langle Arangle is p-nilpotent. Moreover, under the second assumption, it turns out that A=B. This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups.

Finite groups whose commuting conjugacy class graphs have isolated vertices

In this paper, we study a graph whose vertices are the non-trivial conjugacy classes of a finite group, and two conjugacy classes C 1 and C 2 are adjacent if and only if we can find and such that x, y commute. In particular, we investigate groups whose corresponding graphs contain isolated vertices. We classify solvable groups with this property and show that if G is non-solvable, then either is almost simple or G is sharply 2-transitive. We determine almost simple groups with sporadic socles and some families of simple groups with this property.

On EPO-groups with a given property on their conjugacy classes

Let G be a finite group and V(G) be the set of all non-central conjugacy classes of G. Consider the conjugacy class graph of G which is defined as follows: its vertex set is the set V(G) and two distinct vertices xG and yG are connected with an edge if o(x) and o(y) are not coprime. In this article, we determine the structure of groups G whose non-trivial elements have prime orders and whose graph does not have any complete subgraph with six vertices.

Coloring invariants of knots and links are often intractable

Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting homomorphisms $\pi_1(S^3\setminus K) \to G$ that send meridians of $K$ to $C$ is almost parsimoniously $\mathsf{\#P}$-complete. This work is a sequel to a previous result by the authors that counting homomorphisms from fundamental groups of integer homology 3-spheres to $G$ is almost parsimoniously $\mathsf{\#P}$-complete. Where we previously used mapping class groups actions on closed, unmarked surfaces, we now use braid group actions.

The Energy of Conjugacy Classes Graphs of Some Order of Alternating Groups

The energy of a graph , is the sum of all absolute values of the eigen values of the adjacency matrix which is indicated by . An adjacency matrix is a square matrix used to represent of finite graph where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. The group of even permutations of a finite set is known as an alternating group . The conjugacy class graph is a graph whose vertices are non-central conjugacy classes of a group , where two vertices are connected if their cardinalities are not coprime. In this paper, the conjugacy class of alternating group of some order for and their energy are computed. The Maple2019 software and Groups, Algorithms, and Programming (GAP) are assisted for computations.

Topological generation of special linear groups

Let C1,…,Ce be noncentral conjugacy classes of the algebraic group G=SLn(k) defined over a sufficiently large field k, and let Ω:=C1×…×Ce. This paper determines necessary and sufficient conditions for the existence of a tuple (x1,…,xe)∈Ω such that 〈x1,…,xe〉 is Zariski dense in G. As a consequence, a new result concerning generic stabilizers in linear representations of algebraic groups is proved, and existing results on random (r,s)-generation of finite groups of Lie type are strengthened.

The invariably generating graph of the alternating and symmetric groups

Abstract Given a finite group G, the invariably generating graph of G is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of G, and two classes are adjacent if and only if they invariably generate G. In this paper, we study this object for alternating and symmetric groups. The main result of the paper states that if we remove the isolated vertices from the graph, the resulting graph is connected and has diameter at most 6.

The conjugacy class graphs of non-abelian 3-groups

A graph is formed by a pair of vertices and edges. It can be related to groups by using the groups’ properties for its vertices and edges. The set of vertices of the graph comprises the elements or sets from the group while the set of edges of the graph is the properties and condition for the graph. A conjugacy class of an element is the set of elements that are conjugated with . Any element of a group , labelled as , is conjugated to if it satisfies for some elements in with its inverse . A conjugacy class graph of a group is defined when its vertex set is the set of non-central conjugacy classes of . Two distinct vertices and are connected by an edge if and only if their cardinalities are not co-prime, which means that the order of the conjugacy classes of and have common factors. Meanwhile, a simple graph is the graph that contains no loop and no multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is adjacent. Moreover, a -group is the group with prime power order. In this paper, the conjugacy class graphs for some non-abelian 3-groups are determined by using the group’s presentations and the definition of conjugacy class graph. There are two classifications of the non-abelian 3-groups which are used in this research. In addition, some properties of the conjugacy class graph such as the chromatic number, the dominating number, and the diameter are computed. A chromatic number is the minimum number of vertices that have the same colours where the adjacent vertices have distinct colours. Besides, a dominating number is the minimum number of vertices that is required to connect all the vertices while a diameter is the longest path between any two vertices. As a result of this research, the conjugacy class graphs of these groups are found to be complete graphs with chromatic number, dominating number and diameter that are equal to eight, one and one, respectively.

Atoroidal dynamics of subgroups of Out(FN)

We show that for any subgroup $H$ of Out($F_N$), either $H$ contains an atoroidal element or a finite index subgroup $H'$ of $H$ fixes a nontrivial conjugacy class in $F_N$. This result is an analog of Ivanov's subgroup theorem for mapping class groups and Handel-Mosher's subgroup theorem for Out($F_N$) in the setting of irreducible elements.

Topological generation of exceptional algebraic groups

Let G be a simple algebraic group over an algebraically closed field k and let C1,…,Ct be non-central conjugacy classes in G. In this paper, we consider the problem of determining whether there exist gi∈Ci such that 〈g1,…,gt〉 is Zariski dense in G. First we establish a general result, which shows that if Ω is an irreducible subvariety of Gt, then the set of tuples in Ω generating a dense subgroup of G is either empty or dense in Ω. In the special case Ω=C1×⋯×Ct, by considering the dimensions of fixed point spaces, we prove that this set is dense when G is an exceptional algebraic group and t⩾5, assuming k is not algebraic over a finite field. In fact, for G=G2 we only need t⩾4 and both of these bounds are best possible. As an application, we show that many faithful representations of exceptional algebraic groups are generically free. We also establish new results on the topological generation of exceptional groups in the special case t=2, which have applications to random generation of finite exceptional groups of Lie type. In particular, we prove a conjecture of Liebeck and Shalev on the random (r,s)-generation of exceptional groups.

On a Graph Related to Subgroups Generated by a Conjugacy Class in Finite Groups

Let G be a group, and let [Formula: see text](G) be a digraph whose vertices are the non-trivial conjugacy classes of G and there is an arc from a vertex C to a vertex C′ if and only if C ≠ C′ and 〈C〉 ⊆ 〈C′〉. In this paper, we characterize finite groups G whose associated digraphs [Formula: see text](G) are oriented trees.

Avoiding conjugacy classes on the 5-letter alphabet

We construct an infinite word w over the 5-letter alphabet such that for every factor f of w of length at least two, there exists a cyclic permutation of f that is not a factor of w. In other words, w does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [Theoret. Comput. Sci. 726 (2018) 1–4].