(i) A subgroup of PFL(d, q) containing PSL(d, q), in its usual 2-transitive representation; (ii) A collineation group of AG(d, q), containing the group ASL(d, q) of collineations generated by elations, in its usual 2-transitive representation on AG(d, q); (iii) A 7 in its 2-transitive representation of degree 15; (iv) There is a regular normal subgroup, and if x~S then F x acts on S { x } as th...

Let X be a proper CAT(0)-space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is non-elementary and contains a rank-one element then its second bounded cohomology group with coefficients in the regular representation is non-trivial. As a consequence, up to passing to an open subgroup of finite index, either G is a compact extension of a totally disconnected ...

Abstract. In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form H/Γ where Γ is a torsion-free subgroup of minimal index of the congruence two subgroup Γ 2 of the group Γ of positive units of the Lorentzian quadratic form x 1 + · · ·+ x 5 − x 6 . We also show that Γ 2 is a reflection group with respect to a 5-dimensional right-angled convex po...

Let S be a semigroup. For a, x ∈ S such that a = axa, we say that x is an associate of a. A subgroup G of S which contains exactly one associate of each element of S is called an associate subgroup of S. It induces a unary operation in an obvious way, and we speak of a unary semigroup satisfying three simple axioms. A normal cryptogroup S is a completely regular semigroup whose H -relation is a...

the Shimura subgroup Σp of Jo(p), viewed as a subgroup of Jo(p) 2 via the antidiagonal embedding x 7→ (x,−x). Thus we have A = Jo(p)/Σp and, analogously, B = Jo(q) /Σq. Since A and B are known, we consider that to understand A + B is to understand A ∩ B, which is a finite abelian group. The main purpose of this note is to identify A ∩B, up to groups of 2-power order. In other words, we identify...

Let X be a smooth projective curve over C which admits a finite group G of automorphisms. Then G×G acts on the product surface Y = X ×X, and so for each subgroup H ≤ G×G, the quotient Z = H\Y is a normal algebraic surface. Here we shall restrict attention to the case that the subgroup H is the graph of a group automorphism α ∈ Aut(G) of G, i.e. H = ∆α = {(g, α(g)) : g ∈ G}; we propose to call t...

Let G be a finite group and let K = x the conjugacy class of an element G. In this paper, it is proved that if N normal subgroup such coset union ? 1 (the inverse x), then ? ? are solvable. As application, we prove there exists natural number n ? 2 ? ,

Let $G$ be a group. We give an explicit description of the set elements $x \in G$ such that $x^{|G:H|} H$ for every subgroup finite index $H \leqslant G$. This is related to following problem: given two subgroups $H$ and $K$, with index, when does $|HK:H|$ divide $|G:H|$?

A volume invariant is used to characterize those representations of a countable group into a connected semisimple Lie group G which are injective and whose image is a discrete cocompact subgroup of G. Let IT be a discrete cocompact subgroup of G and consider the analytic variety Hom(7r, G) consisting of homomorphisms 0: TT —> G. Denote by K a maximal compact subgroup of G and X = K\G the associ...

Abstract. A subgroup H of a group G is said to be SS-embedded in G if there exists a normal subgroup T of G such that HT is subnormal in G and H T H sG , where H sG is the maximal s- permutable subgroup of G contained in H. We say that a subgroup H is an SS-normal subgroup in G if there exists a normal subgroup T of G such that G = HT and H T H SS , where H SS is an SS-embedded subgroup of ...