Thomas Browning
The commuting probability of a finite group $G$ is the probability that two randomly chosen elements commute. Let $S\subseteq(0,1]$ denote the set of all possible commuting probabilities of finite groups. We prove that $S\cup\{0\}$ is closed, which was conjectured by Keith Joseph in 1977.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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。
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.
The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P \subset (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups.
Let G be a finite group. Let pi be a permutation from S{n}. We study the distribution of probabilities of equality a{1} a{2} ...a{n-1}a{n}=a{pi{1}}^{epsilon{1}} a{pi_{2}}^{epsilon{2}}...a{pi{n-1}}^{epsilon_{n-1}} a_{pi_{n}}^{epsilon{n}}, when pi varies over all the permutations in S{n}, and epsilon{i} varies over the set {+1, -1}. By the paper "Hultman Numbers and Generalized Commuting Probability in Finite Groups" (2017), The case where all epsilon{i} are +1 led to a close connection to Hultman numbers. In this paper we generalize the results, permitting epsilon{i} to be -1. We describe the s…
Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, Martino and Ventura define the 'degree of commutativity' dc_M(G) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if m_n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (m_n) is a…
Commuting probability for approximate subgroups of a finite group
本文研究有限群中近似子群的结构与其交换概率之间的关系，探讨了Pr(A,G)和Pr(A,A)的关联。
A variety of algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with commuting idempotents.
If G is a finite group, then Pr(G) denotes the fraction of ordered pairs of elements of G which commute. We show that, if l \in (2/9,1] is a limit point of the function Pr on finite groups, then l \in \Q and there exists an e = e_l > 0 such that Pr(G) \not\in (l - e_l, l) for any finite group G. These results lend support to some old conjectures of Keith Joseph.
Finite groups with high commuting probability for Sylow subgroups
研究有限群G中，当下中心列项或广义Fitting子群与G的Sylow子群间交换概率较高时，群结构类似于幂零群。
Let $G$ be a finite group and $π$ be a permutation from $S_{n}$. We investigate the distribution of the probabilities of the equality \[ a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}} \] when $π$ varies over all the permutations in $S_{n}$. The probability \[ Pr_π(G)=Pr(a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}}) \] is identical to $Pr_{1}^ω(G)$, with \[ ω=a_{1}a_{2}...a_{n-1}a_{n}a_{π_{1}}^{-1}a_{π_{2}}^{-1}\cdots a_{π_{n-1}}^{-1}a_{π_{n}}^{-1}, \] as it is defined in \cite{DasNath1} and \cite{NathDash1}. The notion of commutativit…
We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over cohe…
We introduce the notion of commuting probability, $p(G)$, for an algebraic group $G$. This notion is inspired by the corresponding notions in finite groups and compact groups. The computation of $p(G)$ for reductive groups is readily done using the notion of $z$-classes. We introduce two generalisations of this relation, $iz$-equivalence and $dz$-equivalence. These notions lead us naturally to the notion of a regular element in $G$. Finally, with the help of this notion of regular elements, we compute $p(G)$ for a connected, linear algebraic group $G$. We also compute the set of limit points o…
It is shown that a wide range of probabilities and limiting probabilities in finite classical groups have integral coefficients when expanded as a power series in 1/q. Moreover it is proved that the coefficients of the limiting probabilities in the general linear and unitary cases are equal modulo 2. The rate of stabilization of the finite dimensional coefficients as the dimension increases is discussed.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。