A Structure Theorem for Small Sumsets in Nonabelian Groups

Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS|< |S|+|H| or |SH| < |S|+|H|. This extends to the nonabelian case classical results for Abelian groups. When we remove the hypothesis |S|<|G|-4|G|^{1/2}+1 we show the existence of counterexamples to the above characterization whose structure is described precisely.Comment: 23 page

Homomorphisms between diffeomorphism groups

For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1- manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if M is any closed manifold, and Diff(M)_0 injects into the diffeomorphism group of a 1-manifold, must M be 1 dimensional? We show that the answer is yes, even under more general conditions. Several lemmas on subgroups of diffeomorphism groups are of independent interest, including results on commuting subgroups and flows.Comment: Contains corrections and additional references. A revised version will appear in Ergodic Theory and Dynamical System

Infinite products of finite simple groups

We classify those sequences $\langle S_{n} \mid n \in \mathbb{N} \rangle$ of finite simple nonabelian groups such that the full product $\prod_{n} S_{n}$ has property (FA).Comment: AMS-LaTex file, 44 pages. To appear in Tran. Amer. Math. So

The Simple Ree groups 2F4(q2){}^2F_4(q^2) are determined by the set of their character degrees

Let $G$ be a finite group. Let ${\rm{cd}}(G)$ be the set of all complex irreducible character degrees of $G.$ In this paper, we will show that if ${\rm{cd}}(G)={\rm{cd}}(H),$ where $H$ is the simple Ree group ${}^2F_4(q^2),q^2\geq 8,$ then $G\cong H\times A,$ where $A$ is an abelian group. This verifies Huppert's Conjecture for the simple Ree groups ${}^2F_4(q^2)$ when $q^2\geq 8.$Comment: 14 pages, to appear in Journal of Algebr

Point regular groups of automorphisms of generalised quadrangles

We study the point regular groups of automorphisms of some of the known generalised quadrangles. In particular we determine all point regular groups of automorphisms of the thick classical generalised quadrangles. We also construct point regular groups of automorphisms of the generalised quadrangle of order $(q-1,q+1)$ obtained by Payne derivation from the classical symplectic quadrangle $\mathsf{W}(3,q)$. For $q=p^f$ with $f\geq 2$ we obtain at least two nonisomorphic groups when $p\geq 5$ and at least three nonisomorphic groups when $p=2$ or $3$. Our groups include nonabelian 2-groups, groups of exponent 9 and nonspecial $p$-groups. We also enumerate all point regular groups of automorphisms of some small generalised quadrangles.Comment: some minor changes (including to title) after referee's comment