Uday Bhaskar Sharma, Anupam Singh
Let $G$ be a finite group. This expository article explores the subject of commuting probability in the group $G$ and its relation with simultaneous conjugacy classes of commuting tuples in $G$. We also point out the relevance of this topic in context of topology where similar problems are studied when $G$ is a Lie group.
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In this paper we compute spectrum, Laplacian spectrum, signless Laplacian spectrum and their corresponding energies of commuting conjugacy class graph of the group $G(p, m, n) = \langle x, y : x^{p^m} = y^{p^n} = [x, y]^p = 1, [x, [x, y]] = [y, [x, y]] = 1\rangle$, where $p$ is any prime, $m \geq 1$ and $n \geq 1$. We derive some consequences along with the fact that commuting conjugacy class graph of $G(p, m, n)$ is super integral. We also compare various energies and determine whether commuting conjugacy class graph of $G(p, m, n)$ is hyperenergetic, L-hyperenergetic or Q-hyperenergetic.
In this paper we compute common neighbourhood (abbreviated as CN) spectrum and energy of commuting conjugacy class graph of several families of finite non-abelian groups. As a consequence of our results we show that the commuting conjugacy class graphs of the groups $D_{2n}$, $T_{4n}$, $SD_{8n}$, $U_{(n,m)}$, $U_{6n}$, $V_{8n}$, $G(p, m, n)$ and some families of groups whose central quotient is isomorphic to $D_{2n}$ or $Z_p \times Z_p$, for some prime $p$, are CN-integral but not CN-hyperenergetic.
For a finite group $G$, we consider the problem of counting simultaneous conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting $n$-tuples in $G$. Let $α_{G,n}$ denote the number of simultaneous conjugacy classes of $n$-tuples, and $β_{G,n}$ the number of simultaneous conjugacy classes of commuting $n$-tuples in $G$. The generating functions $A_G(t) = \sum_{n\geq 0} α_{G,n}t^n,$ and $B_G(t) = \sum_{n\geq 0} β_{G,n}t^n$ are rational functions of $t$. This paper concern studied of normalized functions $A_G(t/|G|)$ and $B_G(t/|G|)$ for finite $p$-groups of rank at most $5…
Commuting Conjugacy Class Graphs of Finite Groups and the Hansen-Vukičević Conjecture
本文计算了有限群交换共轭类图的Zagreb指数，并证明了多种有限群及其商群的交换共轭类图满足Hansen-Vukičević猜想。
The set of all centralizers of elements in a finite group $G$ is denoted by $Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper, the structure of centralizers in a non-abelian finite group $G$ with this property that $\frac{G}{Z(G)} \cong Z_{p^2} \rtimes Z_{p^2}$ is obtained. As a consequence, it is proved that such a group has exactly $[(p+1)^2+1]$ element centralizers and the structure of the commuting conjugacy class graph of $G$ is completely determined.
For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute the genus of $\mathcal{CCC}(G)$ for six well-known classes of non-abelian two-generated groups (viz. $D_{2n}, SD_{8n}, Q_{4m}, V_{8n}, U_{(n, m)}$ and $G(p, m, n)$) and determine whether $\mathcal{CCC}(G)$ for these groups are planar, tor…
The commuting conjugacy class graph of a non-abelian group $G$, denoted by $\mathcal{CCC}(G)$, is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute various spectra and energies of commuting conjugacy class graph of the groups $D_{2n}, Q_{4m}, U_{(n, m)}, V_{8n}$ and $SD_{8n}$. Our computation shows that $\mathcal{CCC}(G)$ is super integral for these groups. We compare vari…
Let $G$ be a finite group. We consider the problem of counting simultaneous conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting $n$-tuples in $G$. Let $α_{G,n}$ denote the number of simultaneous conjugacy classes of $n$-tuples, and $β_{G,n}$ the number of simultaneous conjugacy classes of commuting $n$-tuples in $G$. The generating functions $A_G(t) = \sum_{n\geq 0} α_{G,n}t^n,$ and $B_G(t) = \sum_{n\geq 0} β_{G,n}t^n$ are rational functions of $t$. We show that $A_G(t)$ determines and is completely determined by the class equation of $G$. We show that $α_{G,n}$ gro…
Products of conjugacy classes in $SL_2(R)$
本文针对SL₂(R)中的共轭类乘积问题，提出了计算任意n元组共轭类乘积的方法，解决了该代数结构下共轭类组合运算的计算问题
Spectrum of conjugacy and order super commuting graphs of some finite groups
该研究定义了等价关系下的超图结构，证明其为完全图的广义并，进而推导三类有限群的共轭与阶超交换图的邻接谱和拉普拉斯谱。
Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear grou…
In this article we consider commuting graphs of involution conjugacy classes in the affine Weyl group of type $\tilde C_n$. We show that where the graph is connected the diameter is at most $n+2$.
Forbidden subgraphs on conjugacy class graphs of groups
该文研究有限群的交换与幂零共轭类图的禁用诱导子图性质，刻画了典型群类的对应诱导子图并完成多类群的相关性质分类。
Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G^n, the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in G^n, generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serre's notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We …
A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We prove that the multiple conjugacy problem is solvable between two n-tuples A and B of elements of G whenever the elements of A does not generate an elliptic subgroup. When the edge and vertex groups are infinite cyclic, i.e. G is a Generalized Baumslag-Solitar group, we prove that the multiple conjugacy problem is fully solvable.
Products of conjugacy classes in finite unitary groups $GU(3,q^2)$ and $SU(3,q^2)$
针对有限域上的GU(3)、SU(3)、GL(3)、SL(3)群，解决类乘积问题，给出乘积不含单位矩阵的m元共轭类组完整列表。
In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let $W$ be a classical Weyl group of rank $n$, with $\tilde W$ its corresponding affine Weyl group. Our main result is that if $X$ is a conjugacy class of involutions in $\tilde W$, then the commuting involution graph $\mathcal{C}(\tilde W, X)$ is either disconnected or has diameter at most $n+2$. This bound is known to hold for types $\tilde A_n$ and $\tilde C_n$, so the main work of this paper is to prove the theorem for types $\tilde B_n$ and $\tilde D_n$.
Super commuting graphs of finite groups and their Zagreb indices
本文研究有限群上由相等、共轭和同阶关系定义的超交换图，刻画多类非交换群族的图结构，计算其Zagreb指数并验证Hansen-Vukičević猜想。
Automorphism Groups of the $PSL_2(q)$ Commuting Involution Graphs
本文研究了PSL_2(q)交换involution图的自同构群，通过分析共轭类X的性质，确定了该图的自同构群结构。
Let $H$ be a linear algebraic group over an algebraically closed field of characteristic $p>0$. We prove that any "exponential map" for $H$ induces a bijection between the variety of $r$-tuples of commuting $[p]$-nilpotent elements in $Lie(H)$ and the variety of height $r$ infinitesimal one-parameter subgroups of $H$. In particular, we show that for a connected reductive group $G$ in pretty good characteristic, there is a canonical exponential map for $G$ and hence a canonical bijection between the aforementioned varieties, answering in this case questions raised both implicitly and explicitly…
Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that intersects every conjugacy class of involutions of G.
In this document we study the local connectivity of the sets whose elements are $m$-tuples of pairwise commuting normal matrix contractions. Given $\varepsilon>0$, we prove that there is $δ>0$ such that for any two $m$-tuples of pairwise commuting normal matrix contractions $\mathbf{X}:=(X_1,\ldots,X_m)$ and $\tilde{\mathbf{X}}:=(\tilde{X}_1,\ldots,\tilde{X}_m)$ that are $δ$-close with respect to some suitable distance $ð$ in $(\mathbb{C}^{n\times n})^m$, we can find a $m$-tuple of matrix paths (homotopies) connecting $\mathbf{X}$ to $\mathbf{\tilde{X}}$ relative to the intersection of some $\…
We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure $M$, Polish group $G$ of permutations of $M$, and $n \geq 1$, $G$ has a comeager $n$-diagonal conjugacy class iff the family of all $n$-tuples of $G$-extendable bijections between finitely generated substructures of $M$, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé classes that are not homogenizable. Finally, we investigate $1$- and $2$-diagonal conjugacy cl…
Branching rules and commuting probabilities for Triangular and Unitriangular matrices
本文通过计算分支规则，枚举了奇特征有限域上三角群与单位三角群中小规模情形的同时共轭类，并利用分支矩阵显式求出了相应的交换概率。
The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod the action of the Weyl group, or equivalently to the conjugacy classes of commuting pairs of elements in G. Since the component group for a non-simply connected group is given by some finite dimensional subgroup in the centralizer of an n-tuple, we use diagram automorphisms of the extended Dynkin diagram to prove properties of centralizers of pairs of elemen…
This paper concerns the enumeration of simultaneous conjugacy classes of tuples of commuting unitary matrices and of commuting symplectic matrices over a finite field $\mathbf{F}_q$ of odd size. For any given conjugacy class, the orbits for the action of its centralizer group on itself by conjugation are called branches. We determine the branching rules for the unitary groups $U_2(\mathbf{F}_q), U_3(\mathbf{F}_q)$, and for the symplectic groups $Sp_2(\mathbf{F}_q), Sp_4(\mathbf{F}_q)$.
On a mod $3$ property of $\ell $-tuples of pairwise commuting permutations
本文研究归一化的两两交换置换 $\ell$ 元组计数序列，证明当 $\ell$ 足够大时其关于 $n$ 的对数凹性由 $n$ 模 $3$ 决定，并联系 orbifold 示性数、覆盖计数与群同态计数。
Super graphs on groups, II
本文进一步研究群上九类超图，证明多类共轭超图的普适性，给出其 Schwenk 图复合表示并计算 Wiener 指数。
Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this paper, we seek to extend these techniques to finitely presented non-free groups, with a particular emphasis on polycyclic and metabelian groups that are of interest to non-commutative cryptography. As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using d…
Let $G$ be an algebraic group. For $d\geq 1$, we define the commuting probabilities $cp_d(G) = \frac{dim(\mathfrak C_d(G))}{dim(G^d)}$, where $\mathfrak C_d(G)$ is the variety of commuting $d$-tuples in $G$. We prove that for a reductive group $G$ when $d$ is large, $cp_d(G)\sim \fracα{n}$ where $n=\dim(G)$, and $α$ is the maximal dimension of an Abelian subgroup of $G$. For a finite reductive group $G$ defined over the field $\mathbb F_q$, we show that $cp_{d+1}(G(\mathbb F_q))\sim q^{(α-n)d}$, and give several examples.
For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has a nontrivial conjugacy class inside $N$ of size at most $s-1$. We also extend two results of Barry, MacHale, and N\'ı Shé on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if $d(G)>5/16$ then either $G$ is supersolvable, or $G$ isoclinic to $A_4$, or $G/\Center(G)$ is isoclinic to $A_4$.
Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G. Homotopy properties of the B(q,G) are considered for finite groups. Cohomology calculations are provided for compact Lie groups. The spaces B(2,G) are described in detail for transitively commutative groups.
We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in terms of the extended Dynkin diagram of the simply connected cover, together with the coroot integers and the action of the fundamental group. In the case of three commuting elements, we compute Chern-Simons invariants associated to the corresponding flat bundles over the three-torus, and verify a conjecture of Witten which reveals a surprising symmetry involving the Chern-Simons invariants and the dimensions of the components of the modu…
A survey on conjugacy class graphs of groups
本文综述了群的共轭类图，包括交换、幂零和可解三类，涵盖连通性、图实现、亏格、谱与能量等结果，并列出多个开放问题。
For any (Hausdorff) compact group $G$ with the normalized Haar measure ${\mathbf m}_G$, denote by ${\rm cp}(G)$ the probability ${\mathbf m}_{G\times G}(\{(x,y)\in G\times G \;|\; xy=yx\})$ of commuting a randomly chosen pair of elements of $G$. Here we prove that if ${\rm cp}(G)>0$, then there exists a finite group $H$ such that ${\rm cp}(G)= \frac{{\rm cp}(H)}{|G:F|^2}$, where $F$ is the FC-center of $G$ i.e. the set of all elements of $G$ whose conjugacy classes are finite and $H$ is isoclinic to $F$ with ${\rm cp}(F)={\rm cp}(H)$. The latter equality enables one to transfer many existing r…
Absence of twisting for non-trivial discrete torsion
本文从Dijkgraaf-Witten视角研究n- торus的离散扭力，引入untwisted类子群Br^n(G)，并推导出相关系数精确序列，同时计算了有限子群的分区函数。
Let $(X,μ)$ be a probability space, $G$ a countable amenable group and $(F_n)_n$ a left Følner sequence in $G$. This paper analyzes the non-conventional ergodic averages \[\frac{1}{|F_n|}\sum_{g \in F_n}\prod_{i=1}^d (f_i\circ T_1^g\cdots T_i^g)\] associated to a commuting tuple of $μ$-preserving actions $T_1$, ..., $T_d:G\curvearrowright X$ and $f_1$, ..., $f_d \in L^\infty(μ)$. We prove that these averages always converge in $\|\cdot\|_2$, and that they witness a multiple recurrence phenomenon when $f_1 = \ldots = f_d = 1_A$ for a non-negligible set $A\subseteq X$. This proves a conjecture o…
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