Related papers: Zero-one generation laws for finite simple groups
We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial $f=y^n+\sum_{i=0}^{n-1}a_i(x)y^i\in\mathbb F_q[x][y]$ with i.i.d coefficients $a_i$ taking values in the set…
The goal of this note is to provide yet another proof of the following theorem of Golod: there exists an infinite finitely generated group $G$ such that every element of $G$ has finite order. Our proof is based on the Nielsen-Schreier index…
We call an element of a finite general linear group $ \textrm{GL}(d,q) $ \emph{fat} if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than $d/2$. Fatness of an element can be decided efficiently in practice by…
The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. Their natural analogues are self-similar nil Lie $p$-algebras. In characteristic zero, similar examples of Lie algebras do not exist (Martinez and…
For a finite group $G$, let $\sigma(G)$ be the number of subgroups of $G$ and $\sigma_\iota(G)$ the number of isomorphism types of subgroups of $G$. Let $L=L_r(p^e)$ denote a simple group of Lie type, rank $r$, over a field of order $p^e$…
Every finite simple group can be generated by two elements, and in 2000, Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group every nontrivial element belongs to a generating pair. Groups with…
The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying…
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…
The authors extend to the $q-$tensor square $G \otimes^q G$ of a group $G$, $q$ a non-negative integer, some structural results due to R. D. Blyth, F. Fumagalli and M. Morigi concerning the non-abelian tensor square $G \otimes G$ ($q = 0$).…
We first obtain explicit upper bounds for the proportion of elements in a finite classical group G with a given characteristic polynomial. We use this to complete the proof that the proportion of elements of a finite classical group G which…
In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the 'one and a half generation' property, i.e. when…
Why does Zipf's law give a good description of data from seemingly completely unrelated phenomena? Here it is argued that the reason is that they can all be described as outcomes of a ubiquitous random group division: the elements can be…
We give very precise bounds for the congruence subgroup growth of arithmetic groups. This allows us to determine the subgroup growth of irreducible lattices of semisimple Lie groups. In the most general case our results depend on the…
The classical zero-one law for first-order logic on random graphs says that for any first-order sentence $\phi$ in the theory of graphs, as n approaches infinity, the probability that the random graph G(n, p) satisfies $\phi$ approaches…
We first give a short group theoretic proof of the following result of Lackenby. If $G$ is a large group, $H$ is a finite index subgroup of $G$ admitting an epimorphism onto a non--cyclic free group, and $g$ is an element of $H$, then the…
Let $G$ be a simple algebraic group over an algebraically closed field $k$ and let $C_1, \ldots, C_t$ be non-central conjugacy classes in $G$. In this paper, we consider the problem of determining whether there exist $g_i \in C_i$ such that…
We complete the classification of the finite special linear groups $\SL_n(q)$ which are $(2,3)$-generated, i.e., which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple…
In order to study how well a finite group might be generated by repeated random multiplications, P. Diaconis suggested the following urn model. An urn contains some balls labeled by elements which generate a group G. Two are drawn at random…
Every nonabelian finite simple group of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, has a presentation with a bounded number of generators and relations and total length $O(\log n…
We prove that every non-abelian finite simple group is generated by an involution and an element of prime order.