arXiv subjects
Publications and source records attributed to Gabriel Verret.
Two finite groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. We prove a new sufficient condition for two groups to be compatible. As a corollary, we obtain that nilpotent groups of the same order are compatible, and so are groups of the same square-free order.
Let $\Gamma$ be a finite connected graph and $G$ a vertex-transitive group of its automorphisms. The pair $(\Gamma, G)$ is said to be locally-$L$ if the permutation group induced by the action of the vertex-stabiliser $G_v$ on the set of neighbours of a vertex $v$ in $\Gamma$ is permutation isomorphic to $L$. The maximum growth of $|G_v|$ as a function of $|V\Gamma|$ for locally-$L$ pairs $(\Gamma,G)$ is called the graph growth of $L$. We prove that if $L$ is a transitive permutation group on a set $\Omega$ admitting a nontrivial block $B$ such that the pointwise stabiliser of $\Omega\setminus B$ in $L$ is nontrivial, then the graph growth of $L$ is exponential. This generalises several results in the literature on transitive permutation groups with exponential graph growth.
A classical theorem of Jordan asserts that if a group $G$ acts transitively on a finite set of size at least $2$, then $G$ contains a derangement (a fixed-point free element). Generalisations of Jordan's theorem have been studied extensively, due in part to their applications in graph theory, number theory and topology. We address a generalisation conjectured recently by Ellis and Harper, which says that if $G$ has exactly two orbits and those orbits have equal length $n \geq 2$, then $G$ contains a derangement. We prove this conjecture in the case where $n$ is a product of two primes, and verify it computationally for $n \leq 30$.
Two groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. In this paper, we give new necessary conditions for two groups to be compatible.
We prove that the set of orders of finite quotients of a finitely generated group has natural density 0, 1/2 or 1, and characterise when each of these cases occurs. We apply this to show that the sets of orders of various families of symmetric graphs have natural density 0.
A necessary condition for a Cayley digraph Cay$(R,S)$ to be a regular representation is that there are no non-trivial group automorphisms of $R$ that fix $S$ setwise. A group is DRR-detecting or GRR-detecting if this condition is also sufficient for all Cayley digraphs or graphs on the group, respectively. In this paper, we determine precisely which groups of squarefree order are DRR-detecting, and which are GRR-detecting.
A finite non-regular primitive permutation group $G$ is extremely primitive if a point stabiliser acts primitively on each of its nontrivial orbits. Such groups have been studied for almost a century, finding various applications. The classification of extremely primitive groups was recently completed by Burness and Lee, who relied on an earlier classification of soluble extremely primitive groups by Mann, Praeger and Seress. Unfortunately, there is an inaccuracy in the latter classification. We correct this mistake, and also investigate regular linear spaces which admit groups of automorphisms that are extremely primitive on points.
There is a well known theorem by Deuring which gives a criterion for when the reduction of an elliptic curve with complex multiplication (CM) by the ring of integers of an imaginary quadratic field has ordinary or supersingular reduction. We generalise this and a similar theorem by Goren in dimension 2, and classify the $p$-torsion group scheme of the reduction of 3-dimensional abelian varieties with CM by the ring of integers of a cyclic sextic CM field. We also prove a theorem in arbitrary dimension $g$ that distinguishes ordinary and superspecial reduction for abelian varieties with CM by a cyclic CM field of degree $2g$. As an application, we give algorithms to construct supersingular non-superspecial, and superspecial abelian varieties of dimension 2 (surfaces) and dimension 3, and show that all such varieties have non-integer endomorphisms of small degree.
A graph is called $(k,t)$-regular if it is $k$-regular and the induced subgraph on the neighbourhood of every vertex is $t$-regular. We find new conditions on $(k,t)$ for the existence of such graphs and provide a wide range of examples.
In this paper, we show that every finite simple group of order at least $5$ admits an oriented regular representation of out-valency $2$.
In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $Γ$, $Γ\not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\langle g \rangle$ of length equal to the order of $g$. Moreover, we prove that in this case either $Γ$ belongs to a well understood family of exceptional graphs or at least $5/12$ of the vertices of $Γ$ belong to a regular orbit of $g$. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms $C$ of a connected $3$-valent vertex-transitive graph $Γ$ in terms of the number of vertices of $Γ$ and the length of a longest orbit of $C$.
It is known that there are precisely three transitive permutation groups of degree $6$ that admit an invariant partition with three parts of size $2$ such that the kernel of the action on the parts has order $4$; these groups are called $A_4(6)$, $S_4(6d)$ and $S_4(6c)$. For each $L\in \{A_4(6), S_4(6d), S_4(6c)\}$, we construct an infinite family of finite connected $6$-valent graphs $\{Γ_n\}_{n\in \mathbb{N}}$ and arc-transitive groups $G_n \le \rm{Aut}(Γ_n)$ such that the permutation group induced by the action of the vertex-stabiliser $(G_n)_v$ on the neighbourhood of a vertex $v$ is permutation isomorphic to $L$, and such that $|(G_n)_v|$ is exponential in $|\rm{V}(Γ_n)|$. These three groups were the only transitive permutation groups of degree at most $7$ for which the existence of such a family was undecided. In the process, we construct an infinite family of cubic $2$-arc-transitive graphs such that the dimension of the $1$-eigenspace over the field of order $2$ of the adjacency matrix of the graph grows linearly with the order of the graph.
We say that a finite group G is "DRR-detecting" if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism phi of G such that phi(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product of two cyclic groups of order p is not DRR-detecting, for every odd prime p. We also show that if G and H are nontrivial groups that admit a digraphical regular representation and either gcd(|G|,|H|) = 1, or H is not DRR-detecting, then the direct product G x H is not DRR-detecting. Some of these results also have analogues for graphical regular representations.
Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half its vertices has maximum degree at least $\sqrt{d}$, where $d$ is the valency of the hypercube. This was generalised by Alon and Zheng who proved that every Cayley graph on an elementary abelian $2$-group has the same property. Very recently, Potechin and Tsang proved an analogous results for Cayley graphs on abelian groups. They also conjectured that all Cayley graphs have the analogous property. We disprove this conjecture by constructing various counterexamples, including an infinite family of Cayley graphs of unbounded valency which admit an induced subgraph of maximum valency $1$ on a set of more than half its vertices.
This paper presents a phenomenon which sometimes occurs in tetravalent bipartite locally dart-transitive graphs, called a Base Graph -- Connection Graph dissection. In this dissection, each white vertex is split into two vertices of valence 2 so that the connected components of the result are isomorphic. Given the Base Graph whose subdivision is isomorphic to each component, and the Connection Graph, which describes how the components overlap, we can, in some cases, provide a construction which can make a graph having such a decomposition. This paper investigates the general phenomenon as well as the special cases in which the connection graph has no more than one edge.
The distinguishing number of a graph $G$ is the smallest $k$ such that $G$ admits a $k$-colouring for which the only colour-preserving automorphism of $G$ is the identity. We determine the distinguishing number of finite $4$-valent vertex-transitive graphs. We show that, apart from one infinite family and finitely many examples, they all have distinguishing number $2$.
We determine the automorphism group of the zero-divisor digraph of the semiring of matrices over an antinegative commutative semiring with a finite number of zero-divisors.
Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free in $G$. In this paper, we classify all examples in which $B$ is monolithic (meaning that it has a unique minimal normal subgroup, and that subgroup is not abelian) and core-free in $G$. As a consequence, we obtain a classification of all proper skew morphisms of finite non-abelian simple groups.