Related papers: Invariable generation of finite classical groups

For an ergodic probability-measure-preserving action $G \curvearrowright (X, \mu)$ of a countable group $G$, we define the Rokhlin entropy $h_G^{\mathrm{Rok}}(X, \mu)$ to be the infimum of the Shannon entropies of countable generating…

We show that all finitely generated free-by-cyclic groups are conjugacy separable: if a finitely generated group $G$ surjects onto $\mathbb{Z}$ with free kernel, then for every pair of non-conjugate elements $g,h\in G$, there exists a…

The generating graph encodes how generating pairs are spread among the elements of a group. For more than ten years it has been conjectured that this graph is connected for every finite group. In this paper, we give evidence supporting this…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$…

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…

Let $A$ be a commutative Noetherian ring, and let $R = A[X]$ be the polynomial ring in an infinite collection $X$ of indeterminates over $A$. Let ${\mathfrak S}_{X}$ be the group of permutations of $X$. The group ${\mathfrak S}_{X}$ acts on…

We establish a new mixing theorem for quasirandom groups (finite groups with no low-dimensional unitary representations) $G$ which, informally speaking, asserts that if $g, x$ are drawn uniformly at random from $G$, then the quadruple…

A finite group $G$ is called *uniformly generated*, if whenever there is a (strictly ascending) chain of subgroups $1<\langle x_1\rangle<\langle x_1,x_2\rangle <\cdots<\langle x_1,x_2,\dots,x_d\rangle=G$, then $d$ is the minimal number of…

In this paper, we inspect a relatively unexplored notion of finite generation in semirings, namely semirings in which all congruences are finitely generated. Such semirings are dubbed Congruence Noetherian. After developing sufficient…

Let $d(G)$ be the smallest cardinality of a generating set of a finite group $G.$ We give a complete classification of the finite groups with the property that, whenever $ \langle x_1, \dots, x_{d(G)} \rangle = \langle y_1, \dots, y_{d(G)}…

Let $\mathbb{F}_qG$ be a finite group algebra. We denote by $P(\mathbb{F}_qG)$ the probability that the product of two elements of $\mathbb{F}_qG$ be zero. In this paper, the general formula for computing the $P(\mathbb{F}_qG)$ are…

All groups have 2 generators. For every prime power q, the Generalized Burnside Theorem (Theorem GB) produces an infinite number of solvable groups, Some, such as groups of a prime power exponent, have only elements of finite order and are…

Dixon's famous theorem states that the group generated by two random permutations of a finite set is generically either the whole symmetric group or the alternating group. In the context of random generation of finite groups this means that…

We prove that if $G$ is a finitely generated group and $Z$ is a uniformly recurrent subgroup of $G$ then there exists a minimal system $(X,G)$ with $Z$ as its stability system. This answers a query of Glasner and Weiss \cite{GW} in the case…

Let G be a group and {\phi} be an automorphism of G. Two elements x, y of G are said to be {\phi}-twisted if y = gx{\phi}(g)^{-1} for some g in G. We say that a group G has the R_{\infty}-property if the number of {\phi}-twisted conjugacy…

We show that there exists a family of groups $G_n$ and nontrivial irreducible representations $\rho_n$ such that, for any constant $t$, the average of $\rho_n$ over $t$ uniformly random elements $g_1, \ldots, g_t \in G_n$ has operator norm…

We prove that under two natural probabilistic models (studied by Cleary, Elder, Rechnitzer and Taback), the probability that a random pair of elements of Thompson's group $F$ generate the entire group is positive. We also prove that for any…

In this paper we study the probability that two elements selected at random with replacement from a given finite semigroup act the same by right translation on the semigroup, that is, the chosen elements have the same right matrix.

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by…

Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order $|G|$ of a finite group $G$, then the polynomial invariants of $G$ are generated by polynomials of degrees at most $|G|$.…