Related papers: Finite primitive groups and regular orbits of grou…

Let $G$ be a permutation group on a set $\Omega$, and $k$ a positive integer. The $k$-closure $G^{(k)}$ of $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$, with the same as $G$ orbits of componentwise action on $\Omega^k$. We…

Let $G$ be a finite group, define $I(G)=\{x\in G : x^{2}=1\}$, $C(G)=$ set of the cyclic subgroups of $G$, $i(G)=|I(G)|$ and $c(G)=|C(G)|$. In this article, we will classify finite groups with $i(G)=c(G)-r$ for $r=0,1,$ and $2$. We also…

If $G$ is a group of permutations of a set $\Omega$, then the suborbits of $G$ are the orbits of point-stabilisers $G_\alpha$ acting on $\Omega$. The cardinalities of these suborbits are the subdegrees of $G$. Every infinite primitive…

We prove that the group algebra $KG$ of a group $G$ over a field $K$ is primitive, provided that $G$ has a free subgroup with the same cardinality as $G$, and that $G$ satisfies the following condition $(\ast)$: for each subset $M$ of $G$…

Let $G$ be a finite permutation group on $\Omega,$ a subgroup $K\leqslant G$ is called a fixer if each element in $K$ fixes some element in $\Omega.$ In this paper, we characterize fixers $K$ with $|K|\geqslant |G_\omega|$ for each…

Given a finite group $G$ of order $p^nm$, where $p$ is a prime and $p\nmid m$, we denote by $\psi_p(G)$ the sum of orders of $p$-parts of elements in $G$. In the current note, we prove that $\psi_p(G)\leq\psi_p(C_{p^nm})$, where $C_{p^nm}$…

This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $GL_d(\mathbb{F}_q)$ for which some element $g\in G$ of prime order $r$, in its action on the natural module…

Let $RG$ be the group ring of a finite group $G$ over a commutative ring $R$ with $1$. An element $x$ in $RG$ is said to be skew-symmetric with respect to an involution $\sigma$ of $RG$ if $\sigma(x)=-x.$ A structure theorem for the…

In this article, we introduce the study of a class of finite groups $G$ which admits a subgroup which intersects all non-trivial subgroups of $G$. We also explore a subclass of it consisting of all groups $G$ in which the prime order…

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 finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic…

Let $G$ be a finite solvable permutation group acting faithfully and primitively on a finite set $\Omega$. Let $G_0$ be the stabilizer of a point $\alpha \in \Omega$ The rank of $G$ is defined as the number of orbits of $G_0$ in $\Omega$,…

Let $Q$ be the matrix $\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$ in $GL_2(\mathbb{F}_q)$ where $\mathbb{F}_q$ is a finite field, and let $G$ be the finite cyclic group generated by $Q$. We consider the action of $G$ on the…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

We study definably primitive pseudo-finite permutation groups of finite $SU$-rank. We show that if $(G,X)$ is such a permutation group, then the rank of $G$ can be bounded in terms of the rank of $X$, providing an analogue of a theorem of…

For an element $g$ of a group $G$, a right Engel sink of $g$ is a subset of $G$ containing all sufficiently long commutators $[...[[g ,x],x],\dots ,x]$ for all $x\in G$. A left Engel sink of $g$ is a subset of $G$ containing all…

We say that a class of finite structures for a finite first-order signature is $r$-compressible if each structure $G$ in the class has a first-order description of size at most $O(r(|G|))$. We show that the class of finite simple groups is…

In this paper, we study intersecting sets in primitive and quasiprimitive permutation groups. Let $G \leqslant \mathrm{Sym}(\Omega)$ be a transitive permutation group, and ${S}$ an intersecting set. Previous results show that if $G$ is…

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its…

Let $G$ be a permutation group on a finite set $\Omega$. The base size of $G$ is the minimal size of a subset of $\Omega$ with trivial pointwise stabiliser in $G$. In this paper, we extend earlier work of Fawcett by determining the precise…