Conjugacy Classes Research Articles

Involutions in finite simple groups as products of conjugates

Let G be a finite non-abelian simple group, C a non-identity conjugacy class of G, and Γ C the Cayley graph of G based on C ∪ C − 1 . Our main result shows that in any such graph, there is an involution at bounded distance from the identity.

On the number of centralizers and conjugacy class sizes in finite groups

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 .

A note on a theorem on conjugacy class sizes

A note on a theorem on conjugacy class sizes

Determination of the Conjugacy Classes and Character Table of the Full Non-Rigid Group of Hexachlorocyclopropane Chemical Compound Via Wreath Product of Pair of Permutation Groups

The use of full non-rigid (f.NRG) molecules group theory to study the internal dynamics of molecular structures of chemical compounds is trending in the research space. In this paper, we use computational method to compute the group elements and group table of the Hexachlorocyclopropane molecular (algebraic) structure and thereafter determine the order and conjugacy classes of the group and finally the corresponding symmetry for each permutation group. We considered the point group of the compound which turns out to be isomorphic to the Wreath Products C3wrC2, where Cn denotes a cyclic group of order n. We found that the group has order 18 and 9 conjugacy classes. Investigating the solubility of the components of Hexachlorocyclopropane chemical compound can be considered for further study.

Twisted conjugacy in $\mathrm{SL}_{n}$ and $\mathrm{GL}_{n}$ over subrings of $\overline{\mathbb{F_p}}(t)$

Let \varphi\colon G\to G be an automorphism of an infinite group G . One has an equivalence relation \sim_{\varphi} on G defined as x\sim_{\varphi} y if there exists a z\in G such that y=zx\varphi(z^{-1}) . The equivalence classes are called \varphi -twisted conjugacy classes, and the set G/{\sim}_{\varphi} of equivalence classes is denoted by \mathcal{R}(\varphi) . The cardinality R(\varphi) of \mathcal{R}(\varphi) is called the Reidemeister number of \varphi . We write R(\varphi)=\infty when \mathcal{R}(\varphi) is infinite. We say that G has the R_{\infty} -property if R(\varphi)=\infty for every automorphism \varphi of G . We show that the groups G=\mathrm{GL}_{n}(R), \mathrm{SL}_{n}(R) have the R_{\infty} -property for all n\ge 3 when F[t]\subset R\subsetneq F(t) , where F is a subfield of \overline{\mathbb{F_p}} . When n\ge 4 , we show that any subgroup H\subset \mathrm{GL}_{n}(R) that contains \mathrm{SL}_{n}(R) also has the R_{\infty} -property.

Twisted conjugacy in residually finite groups of finite Prüfer rank

Abstract Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number R ⁢ ( φ ) R(\varphi) (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the R ∞ R_{\infty} property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then R ⁢ ( φ ) R(\varphi) (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map φ ̂ : [ ρ ] ↦ [ ρ ∘ φ ] \hat{\varphi}\colon[\rho]\mapsto[\rho\circ\varphi] (i.e. we prove the TBFT𝑓, the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).

Root cycles in Coxeter groups

Abstract For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ⁢ ( w ) \kappa(w) . Writing w = c 1 ⁢ ⋯ ⁢ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ⁢ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ⁢ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ⁢ ( w ) = max ⁡ Seq κ ⁢ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ⁢ ( X ) = min ⁡ { κ ⁢ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ⁢ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ⁢ ( w ) \kappa(w) and κ ⁢ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ⁢ ( u ) = Seq κ ⁢ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ⁢ ( X ) = κ ⁢ ( u ) = κ ⁢ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .

Groups, Graphs, and Hypergraphs: Average Sizes of Kernels of Generic Matrices with Support Constraints

We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish strong uniformity results pertaining to zeta functions enumerating conjugacy classes of these groups. We deduce that the numbers of conjugacy classes of F q \mathbf {F}_q -points of the groups under consideration depend polynomially on q q . Our approach combines group theory, graph theory, toric geometry, and p p -adic integration. Our uniformity results are in line with a conjecture of Higman on the numbers of conjugacy classes of unitriangular matrix groups. Our findings are, however, in stark contrast to related results by Belkale and Brosnan on the numbers of generic symmetric matrices of given rank associated with graphs.

Row–column duality and combinatorial topological strings

Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat G-bundles for finite groups G in two dimensions, denoted G-TQFTs. We define analogous combinatorial topological strings related to two dimensional topological field theories (TQFTs) based on fusion coefficients of finite groups. These TQFTs are denoted as R(G)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the G-TQFTs and R(G)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory.

On Generic Automorphisms

In this article we investigate generic automorphisms of countable models. Hodges et al. 1993 introduces the notion of SI (small index) generic automorphisms. They used the existence of small index generics to show the small index property of the model. Truss 1989 defines the notion of Truss generic automorphisms. An automorphism f ofM is called Truss generic if its conjugacy class is comeagre in the automorphism group ofM. We study the relationship between these two types of generic automorphisms. We show that either the countable random graph or a countable arithmetically saturated model of True Arithmetic have both SI generic and Truss generic automorphisms. We prove that the dense linear order has the small index property and Truss-generic automorphisms but it does not have SI generic automorphisms. We also construct an example of a countable structure which has SI generics but it does not have Truss generics.

Hurwitz Orbits on Reflection Factorizations of Parabolic Quasi-Coxeter Elements

We prove that two reflection factorizations of a parabolic quasi-Coxeter element in a finite Coxeter group belong to the same Hurwitz orbit if and only if they generate the same subgroup and have the same multiset of conjugacy classes. As a lemma, we classify the finite Coxeter groups for which every reflection generating set that is minimal under inclusion is also of minimum size.

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.

Large N and large representations of Schur line defect correlators

We study the large N and large representation limits of the Schur line defect correlators of the Wilson line operators transforming in the (anti)symmetric, hook and rectangular representations for \U0001d4a9 = 4 U(N) super Yang-Mills theory. By means of the factorization property, the large N correlators of the Wilson line operators in arbitrary representations can be exactly calculated in principle. In the large representation limit they turn out to be expressible in terms of certain infinite series such as Ramanujan’s general theta functions and the q-analogues of multiple zeta values (q-MZVs). Several generating functions for combinatorial objects, including partitions with non-negative cranks and conjugacy classes of general linear groups over finite fields, emerge from the large N correlators. Also we find conjectured properties of the automorphy and the hook-length expansion satisfied by the large N correlators.

Conjugacy Classes in Finitely Generated Groups with Small Cancellation Properties

Conjugacy Classes in Finitely Generated Groups with Small Cancellation Properties

Higher Lie characters and cyclic descent extension on conjugacy classes

Higher Lie characters and cyclic descent extension on conjugacy classes

Applications for the groups S.U.T.(2,p) where p prime upper than 9

The problem of finding the cyclic decomposition (c.d.) for the groups ), where prime upper than 9 is determined in this work. Also, we compute the Artin characters (A.ch.) and Artin indicator (A.ind.) for the same groups, we obtain that after computing the conjugacy classes, cyclic subgroups, the ordinary character table (o.ch.ta.) and the rational valued character table for each group.

Limit pretrees for free group automorphisms: existence

Abstract To any free group automorphism, we associate a real pretree with several nice properties. First, it has a rigid/non-nesting action of the free group with trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism of the real pretree that represents the free group automorphism. Finally and crucially, the loxodromic elements are exactly those whose (conjugacy class) length grows exponentially under iteration of the automorphism; thus, the action on the real pretree is able to detect the growth type of an element. This construction extends the theory of metric trees that has been used to study free group automorphisms. The new idea is that one can equivariantly blow up an isometric action on a real tree with respect to other real trees and get a rigid action on a treelike structure known as a real pretree. Topology plays no role in this construction as all the work is done in the language of pretrees (intervals).

Asymptotic normality of pattern counts in conjugacy classes

Asymptotic normality of pattern counts in conjugacy classes

Improved covering results for conjugacy classes of symmetric groups via hypercontractivity

Abstract We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as normal sets. We show that for any $\epsilon>0$ , there exists $n_0$ such that if $n>n_0$ and A is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon }}$ , then $A^2 \supseteq A_n$ . This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$ . Our proof strategy combines two types of techniques. The first is ‘traditional’ techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck–Shalev, Larsen–Shalev, and more recently, Larsen–Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.

Finite Groups with at Most Nine Non-Central Conjugacy Classes

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