Related papers: Transitive and non-transitive subgroups of permuta…

A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this…

Let $G$ be a finite group and $p$ a fixed prime divisor of $|G|$. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of $G$ is nilpotent or normal or has $p'$-order, then (1)…

In this note, we study non-transitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group $G$, we produce continuum many pairwise non-quasi-isometric regular…

For a non-cyclic finite group $G$, let $\gamma(G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently…

In this article we construct an irreducible simple subgroup G = <q, y, t, w> of GL_{8671}(13) from an irreducible subgroup T of GL_{11}(2) isomorphic to Mathieu's simple group M_{24} by means of Algorithm 2.5 of [13]. We also use the first…

We prove that $\mathrm{SL}_3(\mathbb{Z})$ contains a non-split sharply 2-transitive subgroup, answering a question of Glasner and Gulko. We also prove that $\mathrm{SL}_4(\mathbb{Z})$ contains a non-split sharply 3-transitive subgroup, but…

Let $i(n,k)$ be the proportion of permutations $\pi\in\mathcal{S}_n$ having an invariant set of size $k$. In this note we adapt arguments of the second author to prove that $i(n,k) \asymp k^{-\delta} (1+\log k)^{-3/2}$ uniformly for $1\leq…

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p$, and assume that $p$ is a very good prime for $G$. Let $P$ be a parabolic subgroup whose unipotent radical $U_P$ has nilpotence class less than…

We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either…

A finite non-abelian group $G$ is called commuting integral if the commuting graph of $G$ is integral. In this paper, we show that a finite group is commuting integral if its central factor is isomorphic to ${\mathbb{Z}}_p \times…

A group is $G$ commutative transitive or CT if commuting is transitive on nontrivial elements. A group $G$ is CSA or conjugately separated abelian if maximal abelian subgroups are malnormal. These concepts have played a prominent role in…

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups.…

Let $G$ be an almost simple group with socle $A_n$, the alternating group of degree $n$. We prove that there is a unit of order $pq$ in the integral group ring of $G$ if and only if there is an element of that order in $G$ provided $p$ and…

It is shown that in the units of augmentation one of an integral group ring $\mathbb{Z} G$ of a finite group $G$, a noncyclic subgroup of order $p^{2}$, for some odd prime $p$, exists only if such a subgroup exists in $G$. The corresponding…

Let $m,n$ be positive integers and $p$ a prime. We denote by $\nu(G)$ an extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is a residually finite group satisfying some non-trivial identity $f…

A transitive group $G$ of permutations of a set $\Omega$ is primitive if the only $G$-invariant equivalence relations on $\Omega$ are the trivial and universal relations. If $\alpha \in \Omega$, then the orbits of the stabiliser $G_\alpha$…

Let $H$ and $B$ be subgroups of a finite group $G$ such that $G=N_{G}(H)B$. Then we say that $H$ is \emph{quasipermutable} (respectively \emph{$S$-quasipermutable}) in $G$ provided $H$ permutes with $B$ and with every subgroup (respectively…

The motion of a graph is the minimum number of vertices that are moved by a non-trivial automorphism. Equivalently, it can be defined as the minimal degree of its automorphism group (as a permutation group on the vertices). In this paper we…

We show that if $G$ is any $p$-group of class at most two and exponent $p$, then there exist groups $G_1$ and $G_2$ of class two and exponent $p$ that contain $G$, neither of which can be expressed as a central product, and with $G_1$…

In this short note we present a simple combinatorial trick which can be effectively applied to show the non--existence of sharply transitive sets of permutations in certain finite permutation groups.