Related papers: A note on invariable generation of nonsolvable per…

A subset S of a group G invariably generates G if G = <s^(g(s)) | s in S> for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a…

We say that permutations $\pi_1,\dots, \pi_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $\pi'_1,\dots,\pi'_r$ of these permutations, $\pi'_1,\dots,\pi'_r$ generate $\mathcal{S}_n$. We…

A subset $\left\{x_{1},x_{2},\hdots,x_{d}\right\}$ of a group $G$ \emph{invariably generates} $G$ if $\left\{x_{1}^{g_{1}},x_{2}^{g_{2}},\hdots,x_{d}^{g_{d}}\right\}$ generates $G$ for every $d$-tuple $(g_{1},g_{2}\hdots,g_{d})\in G^{d}$.…

We study the number of random permutations needed to invariably generate the symmetric group, $S_n$, when the distribution of cycle counts has the strong $\alpha$-logarithmic property. The canonical example is the Ewens sampling formula,…

In this article we consider the cycle structure of compositions of pairs of involutions in the symmetric group S_n chosen uniformly at random. These can be modeled as modified 2-regular graphs, giving rise to exponential generating…

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements…

A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response…

A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably…

An involution is a permutation that is its own inverse. Given a permutation $\sigma$ of $[n],$ let $\mathbf{N}_{n}(\sigma)$ denote the number of ways to write $\sigma$ as a product of two involutions of $[n].$ If we endow the symmetric…

We study the indecomposable summands of the permutation module obtained by inducing the trivial $\mathbb{F}(S_a\wr S_n)$-module to the full symmetric group $S_{an}$ for any field $\mathbb{F}$ of odd prime characteristic $p$ such that…

We construct a resolution of irreducible complex representations of the symmetric group $S_n$ by restrictions of representations of $GL_n(\mathbb{C})$ (where $S_n$ is the subgroup of permutation matrices). This categorifies a recent result…

Let $G$ be an almost simple group. We prove that if $x \in G$ has prime order $p \ge 5$, then there exists an involution $y$ such that $<x,y>$ is not solvable. Also, if $x$ is an involution then there exist three conjugates of $x$ that…

A group $G$ is invariably generated if there exists a subset $S \subseteq G$ such that, for every choice $g_s \in G$ for $s \in S$, the group $G$ is generated by $\{ s^{g_s} \mid s \in S \}$. In [GGJ16] Gelander, Golan and Juschenko showed…

A subset $S$ of a group $G$ invariably generates $G$ if $G$ is generated by $\{ s^g(s) | s\in S\} $ for any choice of $g(s)\in G, s\in S$. In case $G$ is topological one defines similarly the notion of topological invariable generation. A…

For a fixed prime power $q$ and natural number $d$ we consider a random polynomial $$f=x^n+a_{n-1}(t)x^{n-1}+\ldots+a_1(t)x+a_0(t)\in\mathbb F_q[t][x]$$ with $a_i$ drawn uniformly and independently at random from the set of all polynomials…

We prove a conjecture dating back to a 1978 paper of D.R.\ Musser~\cite{musserirred}, namely that four random permutations in the symmetric group $\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > \epsilon$ for some…

The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices (n points). In this paper, we generalize many aspects of this situation. We introduce random shifts of…

We propose a framework to study models of computation of indeterministic data, represented by abstract "distributions". In these distributions, probabilities are replaced by "amplitudes" drawn from a fixed semi-ring $S$, of which the…

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.

We show that all permutations in $S_n$ can be generated by affine unicritical polynomials. We use the $\operatorname{PGL}$ group structure to compute the cycle structure of permutations with low Carlitz rank. The tree structure of the group…