Intersecting Families in the Alternating Group and Direct Product of Symmetric Groups

The Erdős-Ko-Rado property for some permutation groups

A subset in a groupG ≤ Sym(n) is intersecting if for any pair of permutations π, σ in the subset there is an i ∈ {1, 2, . . . , n} such that π(i) = σ(i). If the stabilizer of a point is the largest intersecting set in a group, we say that the group has the Erdős-Ko-Rado (EKR) property. Moreover, the group has the strict EKR property if every intersecting set of maximum size in the group is the ...

Beauville Surfaces, Moduli Spaces and Finite Groups

In this paper we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either PSL(2, p), or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves.

Finite Groups with p-Sylow Coverings

An analogue to Dixon's theorem for automaton groups

Dixon’s famous theorem states that the group generated by two random permutations of a finite set is generically either the whole symmetric group or the alternating group. In the context of random generation of finite groups this means that it is hopeless to wish for a uniform distribution – or even a non-trivial one – by drawing random permutations and looking at the generated group. Mealy aut...

Sharply $(n-2)$-transitive Sets of Permutations

Let $S_n$ be the symmetric group on the set $[n]={1, 2, ldots, n}$. For $gin S_n$ let $fix(g)$ denote the number of fixed points of $g$. A subset $S$ of $S_n$ is called $t$-emph{transitive} if for any two $t$-tuples $(x_1,x_2,ldots,x_t)$ and $(y_1,y_2,ldots ,y_t)$ of distinct elements of $[n]$, there exists $gin S$ such that $x_{i}^g=y_{i}$ for any $1leq ileq t$ and additionally $S$ is called e...