Let $\mathbb F$ be a finite field of odd order and $a,b\in\mathbb F\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where χ is the extended quadratic character on $\mathbb F$. Let $Q_{a,b}$ be the quasigroup over $\mathbb F$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi(y-x) \geqslant 0$, and $(x,y)\mapsto x+b(y-x)$ if $\chi(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$ if and only if $\{a,b\}= \{\alpha(c),\alpha(d)\}$ for some $\alpha\in \operatorname{Aut}(\mathbb F)$. We also characterize $\operatorname{Aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric. In proving our results, we also characterize the minimal subquasigroups of $Q_{a,b}$.

Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, Glymour and Halvorson.

Let G be a finite transitive group on a set $\Omega $, let $\alpha \in \Omega $, and let $G_{\alpha }$ be the stabilizer of the point $\alpha $ in G. In this paper, we are interested in the proportion

$$ \begin{align*} \frac{|\{\omega\in \Omega\mid \omega \textrm{ lies in a }G_{\alpha}\textrm{-orbit of cardinality at most 2}\}|}{|\Omega|}, \end{align*} $$

that is, the proportion of elements of $\Omega $ lying in a suborbit of cardinality at most 2. We show that, if this proportion is greater than $5/6$, then each element of $\Omega $ lies in a suborbit of cardinality at most 2, 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.

We give a sufficient and necessary condition for a Praeger–Xu graph to be a Cayley graph.

We show that the automorphism groups of right-angled Artin groups whose defining graphs have at least three vertices are not relatively hyperbolic. We then show that the outer automorphism groups are also not relatively hyperbolic, except for a few exceptional cases. In these cases, the outer automorphism groups are virtually isomorphic to either a finite group, an infinite cyclic group or $\mathrm {GL}_2(\mathbb {Z})$.

Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

It has been recently proved that the automorphism group of a minimal subshift with non-superlinear word complexity is virtually $\mathbb {Z}$ [Cyr and Kra. The automorphism group of a shift of linear growth: beyond transitivity. Forum Math. Sigma3 (2015), e5; Donoso et al. On automorphism groups of low complexity subshifts. Ergod. Th. & Dynam. Sys.36(1) (2016), 64–95]. In this article we extend this result to a broader class proving that the automorphism group of a minimal $\mathcal {S}$ -adic subshift of finite alphabet rank is virtually $\mathbb {Z}$ . The proof is based on a fine combinatorial analysis of the asymptotic classes in this type of subshifts, which we prove are a finite number.

Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.

We introduce the notions ‘virtual automorphism group’ of a minimal flow and ‘semiregular flow’ and investigate the relationship between the virtual and actual group of automorphisms.

The study of graph ${{C}^{*}}$-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph ${{C}^{*}}$-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph ${{C}^{*}}$-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

Order three elements in the exceptional groups of type ${{G}_{2}}$ are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras.

Over an algebraically closed field, there are two conjugacy classes of order three elements in ${{G}_{2}}$ in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.

A relational structure is called reversible iff each bijective endomorphism (condensation) of that structure is an automorphism. We show that reversibility is an invariant of some forms of L∞ω −bi-interpretability, implying that the condensation monoids of structures are topologically isomorphic. Applying these results, we prove that, in particular, all orbits of ultrahomogeneous tournaments and reversible ultrahomogeneous m-uniform hypergraphs are reversible relations and that the same holds for the orbits of reversible ultrahomogeneous digraphs definable by formulas which are not R-negative.

Let $G$ be a finite group with $\mathsf{soc}(G)=\text{A}_{c}$ for $c\geq 5$. A characterization of the subgroups with square-free index in $G$ is given. Also, it is shown that a $(G,2)$-arc-transitive graph of square-free order is isomorphic to a complete graph, a complete bipartite graph with a matching deleted or one of $11$ other graphs.

For an odd prime $p$, a $p$-transposition group is a group generated by a set of involutions such that the product of any two has order 2 or $p$. We first classify a family of $(G,2)$-geodesic transitive Cayley graphs ${\rm\Gamma}:=\text{Cay}(T,S)$ where $S$ is a set of involutions and $T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$. In this case, $T$ is either an elementary abelian 2-group or a $p$-transposition group. Then under the further assumption that $G$ acts quasiprimitively on the vertex set of ${\rm\Gamma}$, we prove that: (1) if ${\rm\Gamma}$ is not $(G,2)$-arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if $T$ is a $p$-transposition group and $S$ is a conjugacy class, then $p=3$ and ${\rm\Gamma}$ is $(G,2)$-arc transitive.

We extend Ahlbrandt and Ziegler’s reconstruction results ([1]) to the metric setting: we show that separably categorical structures are determined, up to bi-interpretability, by their automorphism groups.

Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.

If ${\cal M},{\cal N}$ are countable, arithmetically saturated models of Peano Arithmetic and ${\rm{Aut}}\left( {\cal M} \right) \cong {\rm{Aut}}\left( {\cal N} \right)$, then the Turing-jumps of ${\rm{Th}}\left( {\cal M} \right)$ and ${\rm{Th}}\left( {\cal N} \right)$ are recursively equivalent.