Related papers: On primitive $2$-closed permutation groups of rank…

A finite transitive permutation group is elusive if it contains no derangements of prime order. These groups are closely related to a longstanding open problem in algebraic graph theory known as the Polycirculant Conjecture, which asserts…

We prove that if $L=\mbox{}^2F_4(2^{2n+1})'$ and $x$ is a nonidentity automorphism of $L$ then $G=\langle L,x\rangle$ has four elements conjugate to $x$ that generate $G$. This result is used to study the following conjecture about the…

For a finite non cyclic group $G$, let $\gamma(G)$ be the smallest integer $k$ such that $G$ contains $k$ proper subgroups $H_1,\dots,H_k$ with the property that every element of $G$ is contained in $H_i^g$ for some $i \in \{1,\dots,k\}$…

The pairs $(\mathcal{D},G)$, where $\mathcal{D}$ is a non-trivial $2$-$(k^{2},k,\lambda )$ design, with $\lambda \mid k$, and $G$ is a flag-transitive automorphism group of $\mathcal{D}$ of affine type such that $G \nleq A \Gamma…

We study finite p-groups G of coclass upto 4 for which the group Aut_z(G) of all central automorphisms of G is of minimal possible order. As a consequence, we obtain very short and elementary proofs of main results of Sharma and Gumber [7].

A nut graph is a simple graph of order 2 or more for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry (i.e. are full). It is shown by construction that every finite group…

A graph is said to be $k$-{\em isoregular} if any two vertex subsets of cardinality at most $k$, that induce subgraphs of the same isomorphism type, have the same number of neighbors. It is shown that no $3$-isoregular bicirculant (and more…

Let $\Omega$ be a set of cardinality $n$, $G$ a permutation group on $\Omega$, and $f:\Omega\to\Omega$ a map which is not a permutation. We say that $G$ synchronizes $f$ if the semigroup $\langle G,f\rangle$ contains a constant map. The…

In earlier papers, we showed a decomposition of 2-diregular digraphs (2-dds) and used it to provide some sufficient conditions for these graphs to be non-Hamiltonian; we also showed a close connection between the permanent and determinant…

We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A_Gamma), for certain right-angled Artin groups A_Gamma. This is achieved by introducing a new class of graphs, called focused graphs, whose…

The present work investigates regular, semiregular, and chiral polytopes of any rank $d\geq 3$, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with…

We classify a large class of small groups of finite Morley rank: $N_\circ^\circ$-groups which are the infinite analogues of Thompson's $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a…

Given a permutation group $G$ on a finite set $\Omega$, let $G^{(k)}$ denote the $k$-closure of $G$, that is, the largest permutation group on $\Omega$ having the same orbits in the induced action on $\Omega^k$ as $G$. Recall that a group…

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, $4$-valent one-regular graphs of order $5p^2$, where $p$ is a prime, are classified

Let $\Gamma$ be a finite simple graph. If for some integer $n\geqslant 4$, $\Gamma$ is a $K_n$-free graph whose complement has an odd cycle of length at least $2n-5$, then we say that $\Gamma$ is an $n$-exact graph. For a finite group $G$,…

Let $\mathcal{L}$ be a centric linking system associated to a saturated fusion system on a finite $p$-group $S$. An automorphism of $\mathcal{L}$ is said to be rigid if it restricts to the identity on the fusion system. An inner rigid…

We construct a connected cubic nonnormal Cayley graph on $\mathrm{A}_{2^m-1}$ for each integer $m\geqslant4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on…

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of…

We give a unified approach to analysing, for each positive integer $s$, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally $s$-arc transitive graphs of diameter at least $s$. A graph…

If $G$ is a group acting on a set $\Omega$ and $\alpha, \beta \in \Omega$, the digraph whose vertex set is $\Omega$ and whose arc set is the orbit $(\alpha, \beta)^G$ is called an {\em orbital digraph} of $G$. Each orbit of the stabiliser…