Related papers: Almost simple groups with socle $L_n(q)$ acting on…

Given an explicit presentation of a reflection group of rank two (or any rank two group for that matter), we give a simple procedure for calculating all its systems of imprimitivity, when viewed as a matrix group over the quaternions. This…

We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple…

In this paper, we show that projective special linear groups $S:=L_3(q)$ with $q$ less than $100$ are uniquely determined by their orders and degree patterns of their prime graphs. Indeed, we prove that if $G$ is a finite group whose order…

Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by 3 elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper we consider subgroups at the next…

Let $G$ be a finite group and $n_p(G)$ the number of Sylow $p$-subgroups of $G$. In this paper, we prove if $n_p(G)<p^2$ then almost all numbers $n_p(G)$ are a power of a prime.

Let $\mathrm{SL}_{n}(\mathbb{Z})$ $(n\geq 3)$ be the special linear group and $M^{r}$ be a closed aspherical manifold. It is proved that when $r<n,$ a group action of $\mathrm{SL}_{n}(\mathbb{Z})$ on $M^{r}$ by homeomorphisms is trivial if…

Suppose that a group $G$ has socle $L$ a simple large-rank classical group. Suppose furthermore that $G$ acts transitively on the set of lines of a linear space $\mathcal{S}$. We prove that, provided $L$ has dimension at least 25, then $G$…

We prove that equivariantly simple invariant singularities can only exist for very few representations of a group of prime order: for real representations and some ``almost, but not quite real'' representations.

A regular bipartite graph $\Gamma$ is called semisymmetric if its full automorphism group $\mathrm{Aut}(\Gamma)$ acts transitively on the edge set but not on the vertex set. For a subgroup $G$ of $\mathrm{Aut}(\Gamma)$ that stabilizes the…

Let $G$ be a transitive permutation group of degree $n$ with point stabiliser $H$ and let $r$ be a prime divisor of $n$. We say that $G$ is $r$-elusive if it does not contain a derangement of order $r$. The problem of determining the…

In this article, we study symmetric $(v, k, \lambda)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle $X$ is a finite simple exceptional group of Lie type. We prove a reduction theorem, severely…

Let $G$ be a finite primitive permutation group on a set $\Omega$ with nontrivial point stabilizer $G_{\alpha}$. We say that $G$ is extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$.…

The paper is devoted to projective Clifford groups of quantum $N$-dimensional systems. Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann-Knill theorem). However,…

For each finite primitive subgroup $G$ of $\operatorname{PGL}_4(\mathbb{C})$, we find all the smooth $G$-invariant quartic surfaces. We also find all the faithful representations in $\operatorname{PGL}_4(\mathbb{C})$ of the smooth quartic…

For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the…

A finite transitive permutation group is said to be 3/2-transitive if all the nontrivial orbits of a point stabilizer have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less…

We classify finite-dimensional nilpotent Lie algebras with $2$-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to $SO_2(\mathbb R)$. This enables one to enlarge the class of nilpotent Lie algebras of…

Let $G$ be a finite almost simple group with socle $G_0$. A (nontrivial) factorization of $G$ is an expression of the form $G=HK$, where the factors $H$ and $K$ are core-free subgroups. There is an extensive literature on factorizations of…

Let $G$ be a nontrivial transitive permutation group on a finite set $\Omega$. An element of $G$ is said to be a derangement if it has no fixed points on $\Omega$. From the orbit counting lemma, it follows that $G$ contains a derangement,…

We classify the almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0 \ltimes_A \mathbb R^{2n-1}$ admitting complex or symplectic structures. The matrix $A\in M(2n-1,\mathbb R )$ encodes the adjoint action of $e_0$ on the abelian ideal…