A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let $𝒮$ be a nontrivial finite regular linear space and $G\le \mathrm{Aut}\left(𝒮\right).$ Suppose that $G$ is extremely primitive on points and let rank$\left(G\right)$ be the rank of $G$ on points. We prove that rank$\left(G\right)\ge 4$ with few exceptions. Moreover, we show that $\mathrm{Soc}\left(G\right)$ is neither a sporadic group nor an alternating group, and $G=\mathrm{PSL}(2,q)$ with $q+1$ a Fermat prime if $\mathrm{Soc}\left(G\right)$ is a finite classical simple group.

Let $k$ be a number field, ${𝒪}_k$ its ring of integers, and $f(t,X)\in k\left(t\right)\left[X\right]$ be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations $t\mapsto \overline{t}\in {𝒪}_k$ such that $f(\overline{t},X)$ is still irreducible. In this paper we study the set ${\mathrm{Red}}_f\left({𝒪}_k\right)$ of those $\overline{t}\in {𝒪}_k$ with $f(\overline{t},X)$ reducible. We show that ${\mathrm{Red}}_f\left({𝒪}_k\right)$ is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group...

Nous démontrons que dans la catégorie $ℱ$ des foncteurs entre espaces vectoriels sur ${𝔽}_2$, le produit tensoriel entre le second foncteur injectif standard non constant $V\mapsto {𝔽}_2^{{\left(V^{*}\right)}^{\oplus 2}}$ et un foncteur puissance extérieure est artinien. Seul était antérieurement connu le caractère artinien de cet injectif ; notre résultat constitue une étape pour l’étude du troisième foncteur injectif standard non constant de $ℱ$.Nous utilisons le foncteur de division par le foncteur identité et des considérations issues de la théorie...

We show that if G is a non-archimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable ω-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation...

We consider the full group [φ] and topological full group [[φ]] of a Cantor minimal system (X,φ). We prove that the commutator subgroups D([φ]) and D([[φ]]) are simple and show that the groups D([φ]) and D([[φ]]) completely determine the class of orbit equivalence and flip conjugacy of φ, respectively. These results improve the classification found in [GPS]. As a corollary of the technique used, we establish the fact that φ can be written as a product of three involutions from [φ].

A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by Avelino and Santos (2012, 2013, 2014 and 2015). In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one will be addressed....

Let $A$ be finite dimensional $\mathbf{C}$-algebra which is a complete intersection, i.e. $A=\mathbf{C}[X_1,...,X_n]/(f_1,...,f_n)$ whith a regular sequences $f_1,...,f_n$. Steve Halperin conjectured that the connected component of the automorphism group of such an algebra $A$ is solvable. We prove this in case $A$ is in addition graded and generated by elements of degree 1.