An explicit upper bound on the number of subgroups of a finite group
In this paper we prove that a finite group of order $r$ has at most $$ 7.3722\cdot r^{\frac{\log_2r}{4}+1.5315}$$ subgroups.
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arXiv subjects
Pablo Spiga
Publications and source records attributed to Pablo Spiga.
Normal $2$-coverings of the finite simple groups and their generalizations
Given a finite group $G$, we say that $G$ has weak normal covering number $γ_w(G)$ if $γ_w(G)$ is the smallest integer with $G$ admitting proper subgroups $H_1,\ldots,H_{γ_w(G)}$ such that each element of $G$ has a conjugate in $H_i$, for some $i\in \{1,\ldots,γ_w(G)\}$, via an element in the automorphism group of $G$. We prove that the weak normal covering number of every non-abelian simple group is at least $2$ and we classify the non-abelian simple groups attaining $2$. As an application, we classify the non-abelian simple groups having normal covering number $2$. We also show that the weak normal covering number of an almost simple group is at least two up to one exception. We determine the weak normal covering number and the normal covering number of the almost simple groups having socle a sporadic simple group. Using similar methods we find the clique number of the invariably generating graph of the almost simple groups having socle a sporadic simple group.
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A generalization of Szep's conjecture for almost simple groups
We prove a natural generalization of Szep's conjecture. Given an almost simple group $G$ with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf N}_G (\langle x\rangle){\bf N}_G(\langle y\rangle)$, where we are denoting by ${\bf N}_G(\langle x\rangle)$ and by ${\bf N}_G(\langle y\rangle)$ the normalizers of the cyclic subgroups $\langle x\rangle$ and $\langle y\rangle$. As a consequence of this result, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf C}_G(x){\bf C}_G(y)$.
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A classification of finite primitive IBIS groups with alternating socle
Let $G$ be a finite permutation group on $Ω$. An ordered sequence $(ω_1,\ldots,ω_\ell)$ of elements of $Ω$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups $G$ to the case that the socle of $G$ is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socle an alternating group. Moreover, we propose a conjecture aiming to give a classification of all almost simple primitive IBIS groups.
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The Engel graph of almost simple groups
Given a finite group $G$, the Engel graph of $G$ is a directed graph encoding pairs of elements satisfying some Engel word. From the work of Detomi, Lucchini and Nemmi, the strongly connectivity of the Engel graph of an arbitrary group $G$ is reduced to the understanding of the strongly connectivity of the Engel graph of non-abelian simple groups. In this paper, we investigate the strongly connectivity of the Engel graph of finite non-abelian simple groups.
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On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency
We prove that, if $Γ$ is a finite connected $3$-valent vertex-transitive, or $4$-valent vertex- and edge-transitive graph, then either $Γ$ is part of a well-understood family of graphs, or every non-identity automorphism of $Γ$ fixes at most $1/3$ of the edges. This answers a question proposed by Primož Potočnik and the third author.
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On the Cayleyness of Praeger-Xu graphs
We give a sufficient and necessary condition for a Praeger-Xu graph to be a Cayley graph.
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Finite transitive groups having many suborbits of cardinality at most two and an application to the enumeration of Cayley graphs
Let $G$ be a finite transitive group on a set $Ω$, let $α\in Ω$ and let $G_α$ be the stabilizer of the point $α$ in $G$. In this paper, we are interested in the proportion $$\frac{|\{ω\in Ω\mid ω\textrm{ lies in a }G_α\textrm{-orbit of cardinality at most two}\}|}{|Ω|},$$ that is, the proportion of elements of $Ω$ lying in a suborbit of cardinality at most two. We show that, if this proportion is greater than $5/6$, then each element of $Ω$ lies in a suborbit of cardinality at most two and hence $G$ is classified by a result of Bergman and Lenstra. We also classify the permutation groups attaining the bound $5/6$. We use these results to answer a question concerning the enumeration of Cayley graphs. Given a transitive group $G$ containing a regular subgroup $R$, we determine an upper bound on the number of Cayley graphs on $R$ containing $G$ in their automorphism groups.
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On $n$-partite digraphical representations of finite groups
A group $G$ admits an \textbf{\em $n$-partite digraphical representation} if there exists a regular $n$-partite digraph $Γ$ such that the automorphism group $\mathrm{Aut}(Γ)$ of $Γ$ satisfies the following properties: $\mathrm{Aut}(Γ)$ is isomorphic to $G$, $\mathrm{Aut}(Γ)$ acts semiregularly on the vertices of $Γ$ and the orbits of $\mathrm{Aut}(Γ)$ on the vertex set of $Γ$ form a partition into $n$ parts giving a structure of $n$-partite digraph to $Γ$. In this paper, for every positive integer $n$, we classify the finite groups admitting an $n$-partite digraphical representation.
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Cherlin's conjecture on finite primitive binary permutation groups
A permutation group is {\it binary} if its orbits on $k$-tuples, for any integer $k\geq 2$, can be deduced from its orbits on $2$-tuples. Cherlin conjectured that a finite primitive binary permutation group $G$ must lie in one of three known families. In this paper we complete the proof of this conjecture. To do this we study the case where the group $G$ is almost simple of Lie type.
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A subexponential bound on the cardinality of abelian quotients in finite transitive groups
We show that, for every transitive group $G$ of degree $n\ge 2$, the largest abelian quotient of $G$ has cardinality at most $4^{n/\sqrt{\log_2 n}}$. This gives a positive answer to a 1989 outstanding question of László Kovács and Cheryl Praeger.
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On the height and relational complexity of a finite permutation group
Let $G$ be a permutation group on a set $Ω$ of size $t$. We say that $Λ\subseteqΩ$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $Λ$. We define the height of $G$ to be the maximum size of an independent set, and we denote this quantity $\mathrm{H}(G)$. In this paper we study $\mathrm{H}(G)$ for the case when $G$ is primitive. Our main result asserts that either $\mathrm{H}(G)< 9\log t$, or else $G$ is in a particular well-studied family (the "primitive large--base groups"). An immediate corollary of this result is a characterization of primitive permutation groups with large "relational complexity", the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\mathrm{I}(G)$, the maximum length of an irredundant base of $G$, in which case we prove that if $G$ is primitive, then either $\mathrm{I}(G)<7\log t$ or else, again, $G$ is in a particular family (which includes the primitive large--base groups as well as some others).
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Hypermaps over non-abelian simple groups and strongly symmetric generating sets
A generating pair $x, y$ for a group $G$ is said to be \textbf{\textit{symmetric}} if there exists an automorphism $φ_{x,y}$ of $G$ inverting both $x$ and $y$, that is, $x^{φ_{x,y}}=x^{-1}$ and $y^{φ_{x,y}}=y^{-1}$. Similarly, a group $G$ is said to be \textbf{\textit{strongly symmetric}} if $G$ can be generated with two elements and if all generating pairs of $G$ are symmetric. In this paper we classify the finite strongly symmetric non-abelian simple groups. Combinatorially, these are the finite non-abelian simple groups $G$ such that every orientably regular hypermap with monodromy group $G$ is reflexible.
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A generalization of Sims conjecture for finite primitive groups and two point stabilizers in primitive groups
In this paper we propose a refinement of Sims conjecture concerning the cardinality of the point stabilizers in finite primitive groups and we make some progress towards this refinement. In this process, when dealing with primitive groups of diagonal type, we construct a finite primitive group $G$ on $Ω$ and two distinct points $α,β\in Ω$ with $G_{αβ}\unlhd G_α$ and $G_{αβ}\ne 1$, where $G_α$ is the stabilizer of $α$ in $G$ and $G_{αβ}$ is the stabilizer of $α$ and $β$ in $G$. In particular, this example gives an answer to a question raised independently by Peter Cameron and by Alexander Fomin.
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On the base size of the symmetric and the alternating group acting on partitions
Given three positive integers $n,a,b$ with $n=ab$, we determine the base size of the symmetric group and of the alternating group of degree $n$ in their action on the set of partitions into $b$ parts having cardinality $a$.
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Independent sets of generators of prime power order
A subset $X$ of a finite group $G$ is said to be prime-power-independent if each element in $X$ has prime power order and there is no proper subset $Y$ of $X$ with $\langle Y, Φ(G)\rangle = \langle X, Φ(G)\rangle$, where $Φ(G)$ is the Frattini subgroup of $G$. A group $G$ is $\mathcal{B}_{pp}$ if all prime-power-independent generating sets for $G$ have the same cardinality. We prove that, if $G$ is $\mathcal{B}_{pp}$, then $G$ is solvable. Pivoting on some recent results of Krempa and Stocka, this yields a complete classification of $\mathcal{B}_{pp}$-groups.
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On fixity of arc-transitive graphs
The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation group on the vertex-set of the graph. We prove in this paper that the relative fixity of connected $2$-arc-transitive graphs of a fixed valence tends to $0$ as the number of vertices grows to infinity. We prove the same result for the class of arc-transitive graphs of a fixed prime valence, and more generally, for any class of arc-transitive locally-$L$ graphs, where $L$ is a fixed quasiprimitive graph-restrictive permutation group.
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Constructing infinitely many half-arc-transitive covers of tetravalent graphs
We prove that, given a finite graph $Σ$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $Σ$. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs.
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