Related papers: Classifying Primitive Solvable Permutation Groups …

Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$, which is the maximum number of fixed points of an involution. In this paper we study the involution fixity of almost simple…

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…

We give examples of groups G such that G^00 is different from G^000. We also prove that for groups G definable in an o-minimal structure, G has a "bounded orbit" iff G is definably amenable. These results answer questions of Gismatullin,…

Let $p$ be a prime and $F$ be a finite field of characteristic $p$. Suppose that $FG$ is the group algebra of the finite $p$-group $G$ over the field $F$. Let $V(FG)$ denote the group of normalized units in $FG$ and let $V_*(FG)$ denote the…

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…

The complex representation rings of finite groups are the fundamental class of fusion rings, categorified by the corresponding fusion categories of complex representations. The category of $\mathbb{Z}_+$-modules of finite rank over such a…

Let $G$ be a transitive permutation group acting on a finite set $\Omega$ with $|\Omega|\geqslant 2$. An element of $G$ is said to be a derangement if it has no fixed points on $\Omega$, and by a theorem of Jordan from 1872, $G$ always…

Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to…

For a finite group $G$, the prime graph $\Gamma(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ such that two vertices $p$ and $q$ share an edge if and only if there…

We consider orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Sigma)$, where $\Sigma$ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable…

Let $G\leq{\rm Sym}(\Omega)$ be transitive. Then $G$ is called \textit{elusive} on $\Omega$ if it has no fixed point free element of prime order. The \textit{$2$-closure} of $G$, denoted by $G^{(2),\Omega}$, is the largest subgroup of ${\rm…

We classify (possibly non commutative) algebras of low rank over a domain R. We first review results for algebras of rank 2 and for finite-dimensional division algebras over the real numbers. These results motivate us to consider which…

In this paper we describe all group gradings by an arbitrary finite group $G$ on non-simple finite-dimensional superinvolution simple associative superalgebras over an algebraically closed field $F$ of characteristic 0 or coprime to the…

The \emph{normal rank} of a group is the minimal number of elements whose normal closure coincides with the group. We study the relation between the normal rank of a group and its first $\ell^2$-Betti number and conjecture that inequality…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if every chief factor $H/K$ of $G$ is a $\sigma _{i}$-group for some $i=i(H/K)$. A set…

We will describe an algorithm to arrange all the positive and negative integer numbers. This array of numbers permits grouping them in six different Classes, $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, and $\zeta$. Particularly,…

An element $x$ in a finite group $G$ is said to be \textit{vanishing} if some (complex) irreducible character of $G$ takes value $0$ at $x$. In this article, we prove that every non-abelian finite simple group, except $\mathrm{SL}_2(4)$ and…

We describe all closed permutation groups which act on the set of vectors of a countable vector space $V$ over a prime field of odd order and which contain all automorphisms of $V$. In particular, we prove that their number is finite. These…

Let $G$ be a finite group and let $\Irr(G)$ denote the set of irreducible complex characters of $G$. For a normal subgroup $N \trianglelefteq G$ and $\chi \in \Irr(G)$, we say that $\chi$ is \emph{fully ramified} over $N$ if $\chi(g)=0$ for…

We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups)…