Low-Dimensional Affine Synchronizing Groups

European Journal of Combinatorics, 1992

We determine all finite graphs which admit a distance-transitive, primitive, affine automorphism group G such that a point stabilizer in G is an alternating or symmetric group (moduio scalars). This work forms part of a programme to classify all finite distance-transitive graphs.

European Journal of Combinatorics, 1999

This paper is a contribution to the programme to classify finite distance-transitive graphs and their automorphism groups. We classify pairs ( , G) where is a graph and G is an automorphism group of acting distance-transitively and primitively on the vertex set of , subject to the condition that there is a normal elementary abelian subgroup V in G which acts regularly on the vertex set of and the stabilizer G 0 of a vertex (which is a complement to V in G) has a unique non-abelian composition factor isomorphic to one of the 26 sporadic simple groups. There are exactly 10 examples of , all known for a long time.

Let $\Omega$ be a set of cardinality $n$, $G$ a permutation group on $\Omega$, and $f:\Omega\to\Omega$ a map which is not a permutation. We say that $G$ \emph{synchronizes} $f$ if the transformation semigroup $\langle G,f\rangle$ contains a constant map, and that $G$ is a \emph{synchronizing group} if $G$ synchronizes \emph{every} non-permutation. A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it has previously been conjectured that a primitive group synchronizes every non-uniform transformation. The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks $5$ and $6$. In addition we produce graphs whose automorphism groups have approximately $\sqrt{n...

Theoretical Computer Science, 2006

Pin showed that every p-state automaton (p a prime) containing a cyclic permutation and a non-permutation has a synchronizing word of length at most (p − 1) 2 . In this paper we consider permutation automata with the property that adding any non-permutation will lead to a synchronizing word and establish bounds on the lengths of such synchronizing words. In particular, we show that permutation groups whose permutation character over the rationals splits into a sum of only two irreducible characters have the desired property.

Journal of Algebra, 2020

For a finite group G, the Hurwitz space H in r,g (G) is the space of genus g covers of the Riemann sphere with r branch points and the monodromy group G. In this paper, we give a complete list of primitive genus one systems of affine type. That is, we assume that G is a primitive group of affine type. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in H in r,1 (G). Furthermore, we give a new algorithm for computing large braid orbits on Nielsen classes. This algorithm utilizes a correspondence between the components of H in r,1 (G) and H in r,1 (M ), where M is the point stabilizer in G.

Journal of Group Theory, 2000

This paper introduces the notion of orbit coherence in a permutation group. Let G be a group of permutations of a set Ω. Let π(G) be the set of partitions of Ω which arise as the orbit partition of an element of G. The set of partitions of Ω is naturally ordered by refinement, and admits join and meet operations. We say that G is join-coherent if π(G) is join-closed, and meet-coherent if π(G) is meet-closed. Our central theorem states that the centralizer in Sym(Ω) of any permutation g is meet-coherent, and subject to a certain finiteness condition on the orbits of g, also join-coherent. In particular, if Ω is a finite set then the orbit partitions of elements of the centralizer in Sym(Ω) of g form a lattice. A related result states that the intransitive direct product and the imprimitive wreath product of two finite permutation groups are joincoherent if and only if each of the groups is join-coherent. We also classify the groups G such that π(G) is a chain and prove two further theorems classifying the primitive join-coherent groups of finite degree and the join-coherent groups of degree n normalizing a subgroup generated by an n-cycle.

2013

We prove that for each partition $G=A_1\cup\dots\cup A_n$ of an infinite group $G$ there is a cell $A_i$ of the partition and a subset $F\subset G$ of cardinality $|F|\le n$ such that $G=F\cdot(A_iA_i^{-1})^G$ where $A^G=\bigcup_{x\in G}xAx^{-1}$ for a subset $A\subset G$. This implies that for any partition $G=A_1\cup\dots\cup A_n$ of $G$ into inner-invariant sets $A_i=A_i^G$ there is a cell $A_i$ of the partition such that $G=FA_iA_i^{-1}$ for some finite subset $F\subset G$ of cardinality $|F|\le n$.

Glasgow Mathematical Journal, 1991

Geometriae Dedicata, 2018

We apply a construction of G. A. Margulis to show that there exists a free nonabelian properly discontinuous group of affine transformations of R 3 with both linear and translational parts having integer entries and acting on R 3 without fixed points.

Bulletin of the London Mathematical Society, 2006

This paper introduces the concept of orbit-homogeneity of permutation groups: a group G is orbitt-homogeneous if two sets of cardinality t lie in the same orbit of G whenever their intersections with each G-orbit have the same cardinality. For transitive groups, this coincides with the usual notion of t-homogeneity. This concept is also compatible with the idea of partition transitivity introduced by Martin and Sagan. Further, this paper shows that any group generated by orbit-t-homogeneous subgroups is orbitt-homogeneous, and that the condition becomes stronger as t increases up to n/2 , where n is the degree. So any group G has a unique maximal orbit-t-homogeneous subgroup Ω t (G), and Ωt(G) ≤ Ω t−1 (G). Some structural results for orbit-t-homogeneous groups and a number of examples are also given.