On the minimal degree of a transitive permutation group with stabilizer a 2-group

The minimal degree of a permutation group 𝐺 is defined as the minimal number of non-fixed points of a non-trivial element of 𝐺. In this paper, we show that if 𝐺 is a transitive permutation group of degree 𝑛 having no non-trivial normal 2-subgroups such that the stabilizer of a point is a 2-group, then the minimal degree of 𝐺 is at least \frac{2}{3}n. The proof depends on the classification of finite simple groups.

The space of relative orders and a generalization of Morris indicability theorem

We introduce the space of relative orders on a group and show that it is compact whenever the group is finitely generated. We use this to show that if [Formula: see text] is a finitely generated group acting on the line by order preserving homeomorphisms and some stabilizer of a point is a proper and co-amenable subgroup, then [Formula: see text] surjects onto [Formula: see text]. This is a generalization of a theorem of Morris.

Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds

Let Mμ0 denote S2×S2 endowed with a split symplectic form $\mu \sigma \oplus \sigma $ normalized so that μ≥1 and σ(S2)=1. Given a symplectic embedding $\iota :B_{c}\hookrightarrow M^0_{\mu }$ of the standard ball of capacity c∈(0,1) into Mμ0, consider the corresponding symplectic blow-up $\widetilde {M}^0_{\mu ,c}$. In this paper, we study the homotopy type of the symplectomorphism group ${\mathrm {Symp}}(\widetilde {M}^0_{\mu ,c})$ and that of the space $\Im {\mathrm {Emb}}(B_{c},M^0_{\mu })$ of unparametrized symplectic embeddings of Bc into Mμ0. Writing ℓ for the largest integer strictly smaller than μ, and λ∈(0,1] for the difference μ−ℓ, we show that the symplectomorphism group of a blow-up of ‘small’ capacity c<λ is homotopically equivalent to the stabilizer of a point in Symp(Mμ0), while that of a blow-up of ‘large’ capacity c≥λ is homotopically equivalent to the stabilizer of a point in the symplectomorphism group of a non-trivial bundle $\mathbb {C}P^2\#\,\overline {\mathbb {C}P^2}$ obtained by blowing down $\widetilde {M}^0_{\mu ,c}$. It follows that, for c<λ, the space $\Im {\mathrm {Emb}}(B_{c},M^0_{\mu })$ is homotopy equivalent to S2 ×S2, while, for c≥λ, it is not homotopy equivalent to any finite CW-complex. A similar result holds for symplectic ruled manifolds diffeomorphic to $\mathbb {C}P^2\#\,\overline {\mathbb {C}P^2}$. By contrast, we show that the embedding spaces $\Im {\mathrm {Emb}}(B_{c},\mathbb {C}P^{2})$ and $\Im {\mathrm {Emb}}(B_{c_{1}}\sqcup B_{c_{2}},\mathbb {C}P^{2})$, if non-empty, are always homotopy equivalent to the spaces of ordered configurations $F(\mathbb {C}P^{2},1)\simeq \mathbb {C}P^{2}$ and $F(\mathbb {C}P^{2},2)$. Our method relies on the theory of pseudo-holomorphic curves in 4 -manifolds, on the computation of Gromov invariants in rational 4 -manifolds, and on the inflation technique of Lalonde and McDuff.

Automorphism groups of trivial strongly minimal structures

We study automorphism groups of trivial strongly minimal structures. First we give a characterization of structures of bounded valency through their groups of automorphisms. Then we characterize the triplets of groups which can be realized as the automorphism group of a non algebraic component, the subgroup stabilizer of a point and the subgroup of strong automorphisms in a trivial strongly minimal structure, and also we give a reconstruction result. Finally, using HNN extensions we show that any profinite group can be realized as the stabilizer of a point in a strongly minimal structure of bounded valency.

Non-simplicity of locally finite barely transitive groups

We answer the following questions negatively: Does there exist a simple locally finite barely transitive group (LFBT-group)? More precisely we have: There exists no simple LFBT -group. We also deal with the question, whether there exists a LFBT-group G acting on an infinite set Ω so that G is a group of finitary permutations on Ω. Along this direction we prove: If there exists a finitary LFBT-group G, then G is a minimal non-FC p-group. Moreover we prove that: If a stabilizer of a point in a LFBT-group G is abelian, then G is metabelian. Furthermore G is a p-group for some prime p, G/G′ ≅ Cp∞, and G′ is an abelian group of finite exponent.

On doubly transitive permutation groups of degree prime squared plus one

A doubly transitive permutation group of degreep2+ 1, pa prime, is proved to be doubly primitive forp≠ 2. We also show that if such a group is not triply transitive then either it is a normal extension ofP S L(2,p2) or the stabilizer of a point is a rank 3 group.