Related papers: A proof of Pyber's base size conjecture

Let $G$ be a finite $p$-group. We prove that whenever the commuting probability of $G$ is greater than $(2p^2 + p - 2)/p^5$, the unramified Brauer group of the field of $G$-invariant functions is trivial. Equivalently, all relations between…

Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset $P$ of width $2$. This constant is defined as $\delta(P) = \max_{(x, y)\in P^2}\min\{\mathbb{P}(x\prec y),…

In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces…

Let $D>3$, $D\equiv3\;(4)$ be a prime, and let $\mathcal{C}$ be an ideal class in the field $\mathbb{Q}(\sqrt{-D})$. In this article, we give a new proof that $p(D,\mathcal{C})$, the smallest norm of a split prime…

We show that there exists a universal positive constant $\varepsilon_0 > 0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $$ Y(S^4, [g])…

A difference basis with respect to $n$ is a subset $A \subseteq \mathbb{Z}$ such that $A - A \supseteq \{1, \ldots, n\}$. R\'{e}dei and R\'{e}nyi showed that the minimum size of a difference basis with respect to $n$ is $(c+o(1))\sqrt{n}$…

A base for a finite permutation group $G \le \mathrm{Sym}(\Omega)$ is a subset of $\Omega$ with trivial pointwise stabiliser in $G$, and the base size of $G$ is the smallest size of a base for $G$. Motivated by the interest in groups of…

For a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$ denote the smallest integer $\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $|S| \ge \ell$ has a nonempty zero-sum…

For a finite abelian group $(G,+, 0)$ the Harborth constant $\mathsf{g}(G)$ is the smallest integer $k$ such that each squarefree sequence over $G$ of length $k$, equivalently each subset of $G$ of cardinality at least $k$, has a…

Let $V$ be a vector space of dimension $d$ over $F_q$, a finite field of $q$ elements, and let $G \le GL(V) \cong GL_d(q)$ be a linear group. A base of $G$ is a set of vectors whose pointwise stabiliser in $G$ is trivial. We prove that if…

Fix a positive integer $d$ and let $\Gamma_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $\Gamma_d$ were studied by Babai, Cameron and P\'{a}lfy in the 1980s and they determined…

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group and recall that an element in $G$ is a derangement if it has no fixed points on $\Omega$. Let $\Delta(G)$ be the set of derangements in $G$ and define $\delta(G) =…

We prove that a quotient G/N of a subgroup G of Sym(n) by a nonabelian minimal normal subgroup N of G embeds into Sym(m) for some $m<n$. This result was proved previously by Robert Chamberlain, and we also prove that,if G is transitive,…

Given a quantum permutation group $G\subset S_N^+$, with orbits having the same size $K$, we construct a universal matrix model $\pi:C(G)\to M_K(C(X))$, having the property that the images of the standard coordinates $u_{ij}\in C(G)$ are…

We study permutation groups of given minimal degree without the classical primitivity assumption. We provide sharp upper bounds on the order of a permutation group of minimal degree m and on the number of its elements of any given support.…

Let $G$ be a finite abelian group and $p$ be the smallest prime dividing $|G|$. Let $S$ be a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H \subsetneq G$, $S$ contains at most $|H|-1$ terms from $H$. Let…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

L\'evai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all…

The following short note provides an alternative proof of a result of Coornaert: namely, that given a non-elementary word-hyperbolic group $G$ with a finite generating set $X$, there exist constants $\lambda,D > 1$ such that \[…

If $G$ is a finite Abelian group, define $s_{k}(G)$ to be the minimal $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. Recently Bitz et al. proved that if $n = exp(G)$, then…