Related papers: Large totally symmetric sets
A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the…
Totally symmetric sets are a recently introduced tool for studying homomorphisms between groups. In this paper, we give full classifications of totally symmetric sets in certain families of groups and bound their sizes in others. As a…
We survey the theory of totally symmetric sets, with applications to homomorphisms of symmetric groups, braid groups, linear groups, and mapping class groups.
We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in $[0,\frac{1}{3}]$, answering a question of Cameron, and that the number of those…
A group $G$ is said to be totally $k$-closed for a positive integer $k$ if, in each of its faithful permutation representations on a set $\Omega^k$, $G$ is the largest subgroup of the symmetric group $\operatorname{Sym}(\Omega)$ that…
We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order $p$, acting on a finite set of given size $n$, which is sharp for certain $n$ and $p$. Namely, we prove that if $n\equiv k\pmod{p}$…
Let $\mbox{$\cal F$}\subseteq 2^{[n]}$ be a fixed family of subsets. Let $D(\mbox{$\cal F$})$ stand for the following set of Hamming distances: $$ D(\mbox{$\cal F$}):=\{d_H(F,G):~ F, G\in \mbox{$\cal F$},\ F\neq G\}. $$ $\mbox{$\cal F$}$ is…
In this article, we classify all symmetric generalized numerical semigroups in $\mathbb{N}^d$ of embedding dimension $2d+1$. Consequently, we show that in this case the property of being symmetric is equivalent to have a unique maximal gap…
Recently George Bergman proved that the symmetric group of an infinite set possesses the following property which we call by the {\it universality of finite width}: given any generating set $X$ of the symmetric group of an infinite set…
The strong symmetric genus of a group is the smallest genus of a surface on which the group acts faithfully as a group of orientation preserving automorphisms. In this paper we announce and prove the strong symmetric genus for the…
Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of…
A subset $A$ of a group $G$ is called product-free if there is no solution to $a=bc$ with $a,b,c$ all in $A$. It is easy to see that the largest product-free subset of the symmetric group $S_n$ is obtained by taking the set of all odd…
Let $\k$ be a characteristic zero PID, $S$ be a $\k$-algebra and $T\subseteq S$ be a full rank subalgebra. Suppose the algebra $T$ is symmetric. It is important to know when $T$ is a {\em maximal symmetric subalgebra} of $S$, i.e. no…
A perfect matching in a hypergraph is a set of edges that partition the set of vertices. We study the complexity of deciding the existence of a perfect matching in orderable and separable hypergraphs. We show that the class of orderable…
For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant each…
We characterize the ``best'' model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct ``best'' model geometries in an appropriate sense--these are the maximally symmetric trees. The…
An $(n,k)$ sequence covering array is a set of permutations of $[n]$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of at least one of the permutations. An $(n,k)$ sequence covering array is perfect if there is a…
A set of integers is sum-free if it contains no solution to the equation $x+y=z$. We study sum-free subsets of the set of integers $[n]=\{1,\ldots,n\}$ for which the integer $2n+1$ cannot be represented as a sum of their elements. We prove…
The strong symmetric genus of a finite group is the minimum genus of a compact Riemann surface on which the group acts as a group of automorphisms preserving orientation. A characterization of the infinite number of groups with strong…
We formalize symmetry breaking as a set-covering problem. For the case of breaking symmetries on graphs, a permutation covers a graph if applying it to the graph yields a smaller graph in a given order. Canonical graphs are those that…