On the covering number of symmetric groups of even degree

If a group $G$ is the union of proper subgroups $H_1, \dots, H_k$, we say that the collection $\{H_1, \dots H_k \}$ is a cover of $G$, and the size of a minimal cover (supposing one exists) is the covering number of $G$, denoted $\sigma(G)$. Mar\'oti showed that $\sigma(S_n) = 2^{n-1}$ for $n$ odd and sufficiently large, and he also gave asymptotic bounds for $n$ even. In this paper, we determine the exact value of $\sigma(S_n)$ when $n$ is divisible by $6$.Comment: Manuscript has been completely rewritten to fix gaps in proofs and improve readabilit

Graphs that contain multiply transitive matchings

Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$ contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the matching $\mathcal{M}$ setwise allows the edges of $\mathcal{M}$ to be permuted in any fashion. A matching $\mathcal{M}$ is \textit{2-transitive} if the setwise stabilizer of $\mathcal{M}$ in $\text{Aut}(\Gamma)$ can map any ordered pair of distinct edges of $\mathcal{M}$ to any other ordered pair of distinct edges of $\mathcal{M}$. We provide constructions of graphs with a permutable matching; we show that, if $\Gamma$ is an arc-transitive graph that contains a permutable $m$-matching for $m \ge 4$, then the degree of $\Gamma$ is at least $m$; and, when $m$ is sufficiently large, we characterize the locally primitive, arc-transitive graphs of degree $m$ that contain a permutable $m$-matching. Finally, we classify the graphs that have a $2$-transitive perfect matching and also classify graphs that have a permutable perfect matching.Comment: to appear in European Journal of Combinatoric

Transitive PSL(2,11)-invariant k-arcs in PG(4,q)

A \textit{k}-arc in the projective space ${\rm PG}(n,q)$ is a set of $k$ projective points such that no subcollection of $n+1$ points is contained in a hyperplane. In this paper, we construct new $60$-arcs and $110$-arcs in ${\rm PG}(4,q)$ that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set $\mathcal{P}$ of projective points in the projective space of dimension $n$ over an algebraic number field $\mathcal{Q}(\xi)$, determines a complete list of primes $p$ for which the reduction modulo $p$ of $\mathcal{P}$ to the projective space ${\rm PG}(n,p^h)$ may fail to be a $k$-arc. Using these methods, we prove that there are infinitely many primes $p$ such that ${\rm PG}(4,p)$ contains a ${\rm PSL}(2,11)$-invariant $110$-arc, where ${\rm PSL}(2,11)$ is given in one of its natural irreducible representations as a subgroup of ${\rm PGL}(5,p)$. Similarly, we show that there exist ${\rm PSL}(2,11)$-invariant $110$-arcs in ${\rm PG}(4,p^2)$ and ${\rm PSL}(2,11)$-invariant $60$-arcs in ${\rm PG}(4,p)$ for infinitely many primes $p$.Comment: 21 pages; updated and revise

On the number of reachable pairs in a digraph

A pair $(u, v)$ of (not necessarily distinct) vertices in a directed graph $D$ is called a reachable pair if there exists a directed path from $u$ to $v$. We define the weight of $D$ to be the number of reachable pairs of $D$, which equals the sum of the number of vertices in $D$ and the number of directed edges in the transitive closure of $D$. In this paper, we study the set $W(n)$ of possible weights of directed graphs on $n$ labeled vertices. We prove that $W(n)$ can be determined recursively and describe the integers in the set. Moreover, if $b(n) \geqslant n$ is the least integer for which there is no digraph on $n$ vertices with exactly $b(n)+1$ reachable pairs, we determine $b(n)$ exactly through a simple recursive formula and find an explicit function $g(n)$ such that $|b(n)-g(n)| < 2n$ for all $n \geqslant 3$. Using these results, we are able to approximate $|W(n)|$ -- which is quadratic in $n$ -- with an explicit function that is within $30n$ of $|W(n)|$ for all $n \geqslant 3$, thus answering a question of Rao. Since the weight of a directed graph on $n$ vertices corresponds to the number of elements in a preorder on an $n$ element set and the number of containments among the minimal open sets of a topology on an $n$ point space, our theorems are applicable to preorders and topologies.Comment: 36 pages, now including supporting data for calculations; to appear in Australasian Journal of Combinatoric

Fuchs' problem for 2-groups

Nearly $60$ years ago, L\'{a}szl\'{o} Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group of units of a finite ring. In this paper, we consider the question of which $2$-groups are realizable as unit groups of finite rings, a necessary step toward determining which nilpotent groups are realizable. We prove that all $2$-groups of exponent $4$ are realizable in characteristic $2$. Moreover, while some groups of exponent greater than $4$ are realizable as unit groups of rings, we prove that any $2$-group with a self-centralizing element of order $8$ or greater is never realizable in characteristic $2^m$, and consequently any indecomposable, nonabelian group with a self-centralizing element of order $8$ or greater cannot be the group of units of a finite ring.Comment: 19 page

Homomorphisms of matrix algebras and constructions of Butson-Hadamard matrices

An $n \times n$ matrix $H$ is Butson-Hadamard if its entries are $k^{\text{th}}$ roots of unity and it satisfies $HH^* = nI_n$. Write $BH(n, k)$ for the set of such matrices. Suppose that $k = p^{\alpha}q^{\beta}$ where $p$ and $q$ are primes and $\alpha \geq 1$. A recent result of {\"O}sterg{\aa}rd and Paavola uses a matrix $H \in BH(n,pk)$ to construct $H' \in BH(pn, k)$. We simplify the proof of this result and remove the restriction on the number of prime divisors of $k$. More precisely, we prove that if $k = mt$, and each prime divisor of $k$ divides $t$, then we can construct a matrix $H' \in BH(mn, t)$ from any $H \in BH(n,k)$.Comment: 5 page

A note on relative hemisystems of Hermitian generalised quadrangles

In this paper we introduce a set of sufficient criteria for the construction of relative hemisystems of the Hermitian space $\mathrm{H}(3,q^2)$, unifying all known infinite families. We use these conditions to provide new proofs of the existence of the known infinite families of relative hemisystems. Reproving these results has allowed us to find new relative hemisystems closely related to an infinite family of Cossidente's, and develop techniques that are likely to be useful in finding relative hemisystems in future

On 22-arc-transitive graphs of order kpnkp^n

We show that there exist functions $c$ and $g$ such that, if $k$, $n$ and $d$ are positive integers with $d> g(n)$ and $\Gamma$ is a $d$-valent $2$-arc-transitive graph of order $kp^n$ with $p$ a prime, then $p\leqslant kc(d)$. In other words, there are only finitely many $d$-valent 2-arc-transitive graphs of order $kp^n$ with $d>g(n)$ and $p$ prime. This generalises a recent result of Conder, Li and Poto\v{c}nik.Comment: Fixed a mistak

A classification of finite antiflag-transitive generalized quadrangles

A generalized quadrangle is a point-line incidence geometry $\mathcal{Q}$ such that: (i) any two points lie on at most one line, and (ii) given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. The finite Moufang generalized quadrangles were classified by Fong and Seitz (1973), and we study a larger class of generalized quadrangles: the \emph{antiflag-transitive} quadrangles. An antiflag of a generalized quadrangle is a non-incident point-line pair $(P, \ell)$, and we say that the generalized quadrangle $\mathcal{Q}$ is antiflag-transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle $\mathcal{Q}$ is antiflag-transitive, then $\mathcal{Q}$ is either a classical generalized quadrangle or is the unique generalized quadrangle of order $(3,5)$ or its dual

Spectra of Hadamard matrices

A Butson Hadamard matrix $H$ has entries in the kth roots of unity, and satisfies the matrix equation $HH^{\ast} = nI_{n}$. We write $\mathrm{BH}(n, k)$ for the set of such matrices. A complete morphism of Butson matrices is a map $\mathrm{BH}(n, k) \rightarrow \mathrm{BH}(m, \ell)$. In this paper, we develop a technique for controlling the spectra of certain Hadamard matrices. For each integer $t$, we construct a real Hadamard matrix $H_{t}$ of order $n_{t} = 2^{2^{t-1}-1}$ such that the minimal polynomial of $\frac{1}{\sqrt{n_{t}}}H_{t}$ is the cyclotomic polynomial $\Phi_{2^{t+1}}(x)$. Such matrices yield new examples of complete morphisms $$\mathrm{BH}(n, 2^{t}) \rightarrow \mathrm{BH}(2^{2^{t-1}-1}n, 2)\,,$$ for each $t \geq 2$, generalising a well-known result of Turyn.Comment: 12 page