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Consider a smooth action G × M → M of a compact connected Lie group G on a connected manifold M. Assume the existence of a point of M whose isotropy group has a single element (a free point). Then we prove that there exist two complete vector field X , X 1 such that their group of automorphisms equals G regarded as a group of diffeomorphisms of M (the existence of a free point implies that the action of G is effective). Moreover, some examples of effective actions with no free point where this result fails are exhibited. [ABSTRACT FROM AUTHOR]

Let the pair ( G , M ) R denote a group G acting on an R -module M , R a unitary ring. We ask for the existence of an R -local space X such that ( E ( X ) , π k ( X ) ) R is equivalent, in a natural way, to ( G , M ) R , for some k ≥ 2 , where E ( X ) denotes the group of homotopy classes of self-homotopy equivalences of X . If such an X exists, we say that X realizes the group action ( G , M ) R . We prove that if G is finite and acts faithfully on a finitely generated Q -module M , there exist infinitely many rational spaces realizing ( G , M ) Q . Our proof relies on providing a positive answer to Kahn's realizability problem for a large class of orthogonal groups that strictly contains finite ones. As a matter of fact, we enlarge the class of groups that is known to be realizable in the classical Kahn's sense. [ABSTRACT FROM AUTHOR]

Consider a smooth effective action of a torus Tn on a connected C∞-manifold M. Assume that M is not a torus endowed with the natural action. Then we prove that there exists a complete vector field X on M such that the automorphism group of X equals Tn×R, where the factor R comes from the flow of X and Tn is regarded as a subgroup of Diff(M). Thus one may reconstruct the whole action of Tn from a single vector field. [ABSTRACT FROM AUTHOR]

We provide a nilpotency criterion for fusion systems in terms of the vanishing of its cohomology with twisted coefficients. [ABSTRACT FROM AUTHOR]

An oriented closed connected $N$ -manifold $M$ is inflexible if it does not admit self-maps of unbounded degree. In addition, if all the maps from any other oriented closed connected $N$ -manifold have bounded degree, then $M$ is said to be strongly inflexible. The existence of simply-connected inflexible manifolds was established by Arkowitz and Lupton. However, the existence of simply-connected strongly inflexible manifolds is still an open question. We provide an algorithm relying on Sullivan models that allows us to prove that all, but one, of the known examples of simply-connected inflexible manifolds are not strongly inflexible. [ABSTRACT FROM AUTHOR]

We show that the vertex coloring problem for finite simplicial complexes can be translated into the algebraic homotopy problem of ellipticity for rational spaces. We follow the ideas of Lechuga-Murillo for the classical vertex coloring of graphs. [ABSTRACT FROM AUTHOR]

We prove that the isomorphism type of a large class of groups is determined by the class of differential graded algebras over an integral domain with mild conditions, on which these groups act faithfully. [ABSTRACT FROM PUBLISHER]

We prove that every finite group G can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces X. To construct those spaces we introduce a new technique which leads, for example, to the existence of infinitely many inflexible manifolds. Further applications to representation theory will appear in a separate paper. [ABSTRACT FROM AUTHOR]

Let G be a finite 2-group of maximal nilpotency class, and let BG be its classifying space. We prove that iteratedMassey products in H*(BG; F2) do characterize the homotopy type of BG among 2-complete spaces with the same cohomological structure. As a consequence we get an alternative proof of the modular isomorphism problem for 2-groups of maximal nilpotency class. [ABSTRACT FROM AUTHOR]

This is a matricial description of all the fourteen fine group gradings on the exceptional Lie algebra ο(8, ℂ). [Copyright &y& Elsevier]

Let X be a finite type An2-polyhedron, n ≥ 2. In this paper, we study the quotient group ℰ (X) / ℰ*(X), where ℰ (X) is the group of self-homotopy equivalences of X and ℰ*(X) the subgroup of self-homotopy equivalences inducing the identity on the homology groups of X. We show that not every group can be realised as ℰ (X) or ℰ (X) / ℰ *(X) for X an An2-polyhedron, n ≥ 3, and specific results are obtained for n = 2. [ABSTRACT FROM AUTHOR]

In this short note we prove that a saturated fusion system admitting some special type of automorphism is nilpotent. This generalizes classical results by J. G. Thompson. [ABSTRACT FROM AUTHOR]

In this paper we show that evolution algebras over any given field \Bbbk are universally finite. In other words, given any finite group G, there exist infinitely many regular evolution algebras X such that Aut(X)\cong G. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme \mathbf {G} over \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra. [ABSTRACT FROM AUTHOR]

As it is well known, there is an element in the family of classical Lie algebras which presents more symmetries than the others. We are speaking, of course, about the algebra which, together with the usual two-fold symmetry which is common to the whole family, presents an order three symmetry which makes of it an “exceptional” classical Lie algebra. In this work we describe the fine group gradings induced by outer automorphisms of the Lie algebra . The qualitative approach used here emphasizes some algebraic aspects, as triality, and it is devoted to gradings involving the outer automorphisms of order three. The computation of the remaining fine group gradings, the ones induced by inner automorphisms and outer ones of order two, follows the same pattern as the classical case with n ≥ 5 and so it has been previously considered in the literature. [ABSTRACT FROM AUTHOR]

Abstract: Often a localization functor (in the category of groups) sends a finite simple group to another finite simple group. We study when such a localization also induces a localization between the automorphism groups and between the universal central extensions. As a consequence we exhibit many examples of localizations of finite simple groups which are not simple. [Copyright &y& Elsevier]

In this paper we answer negatively a question posed by Casacuberta, Farjoun, and Libman about the preservation of perfect groups under localization functors. Indeed, we show that a certain P -localization of Berrick and Casacuberta's universal acyclic group is not perfect. We also investigate under which conditions perfectness is preserved: For instance, we show that if the localization of a perfect group is finite then it is perfect. [ABSTRACT FROM AUTHOR]