Related papers: The base size of a primitive diagonal group

If $G$ is a finite primitive complex reflection group, all reflection subgroups of $G$ and their inclusions are determined up to conjugacy. As a consequence, it is shown that if the rank of $G$ is $n$ and if $G$ can be generated by $n$…

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for…

Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving…

We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with…

The covering number of a group $G$, denoted by $\sigma(G)$, is the size of a minimal collection of proper subgroups of $G$ whose union is $G$. We investigate which integers are covering numbers of groups. We determine which integers $129$…

In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is…

In this thesis we study the following property of a finite group G: the minimal number n such that G embeds in Sn. We start with an explicit formula for the number n for abelian groups. Then, we study the behavior of this group property in…

Two fundamental ways to represent a group are as permutations and as matrices. In this paper, we study linear representations of groups that intertwine with a permutation representation. Recently, D'Alconzo and Di Scala investigated how…

Let $G$ be a finite insoluble group with soluble radical $R(G)$. In this paper we investigate the soluble graph of $G$, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in $G…

Let $G$ be a finite group and let $\mathcal{M}$ be a set of maximal subgroups of $G$. We say that $\mathcal{M}$ is irredundant if the intersection of the subgroups in $\mathcal{M}$ is not equal to the intersection of any proper subset. The…

In this short note, it is proved that both the number of primitive characters and the number of quasi-primitive characters in a finite group $G$ is divisible by $|G:G'|$, where $G'$ is the derived subgroup of $G$.

Let $H$ be a subgroup of $\text{Sym}_n$, the symmetric group of degree $n$. For a fixed integer $l \geq 2$, the group $G$ presented with generators $x_1, x_2, \ldots ,x_n$ and with relations $x_{i_1}x_{i_2}\cdots x_{i_l} =x_{\sigma (i_1)}…

Let G be a permutation group acting on {1,...,n}, and < be any admissible term order on the polynomial ring K[x_1,...,x_n]. We prove that the invariant ring K[x_1,...,x_n]^G of G has a finite SAGBI basis if, and only if, G is generated by…

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is…

Let $n$ be a positive integer, $q$ be a prime power, and $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$. Let $G := V \rtimes G_0$, where $G_0$ is an irreducible subgroup of ${\rm GL}(V)$ which is maximal by inclusion with…

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 $(g_{1},g_{2}\hdots,g_{d})\in G^{d}$.…

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are…

Let $G$ be a group such that $G/Z(G)$ is finite and simple. The non-commuting, non-generating graph $\Xi(G)$ of $G$ has vertex set $G \setminus Z(G)$, with edges corresponding to pairs of elements that do not commute and do not generate…

For a set $\Omega$ an unordered relation on $\Omega$ is a family R of subsets of $\Omega.$ If R is such a relation we let G(R) be the group of all permutations on $\Omega$ that preserves R, that is g belongs to G(R) if and only if x in R…