670 notes - OSU Department of Mathematics

... Use the handout. The concept of fundamental groups is, well, fundamental in topology. For example, one of the most important ways of studying a knot in R3 is to study the fundamental group of its complement. (Draw a picture.) Example. I won’t prove these claims, but I hope they appear plausible. A t ...

... Use the handout. The concept of fundamental groups is, well, fundamental in topology. For example, one of the most important ways of studying a knot in R3 is to study the fundamental group of its complement. (Draw a picture.) Example. I won’t prove these claims, but I hope they appear plausible. A t ...

Trees and amenable equivalence relations

... class becomes tree isomorphic to the Cayley graph of Z (with generating set {±1}), i.e., to the tree with two edges belonging to every vertex. In this paper we prove two less immediate results. First, we show, in the presence of a finite invariant measure, that if the equivalence relation is amenabl ...

... class becomes tree isomorphic to the Cayley graph of Z (with generating set {±1}), i.e., to the tree with two edges belonging to every vertex. In this paper we prove two less immediate results. First, we show, in the presence of a finite invariant measure, that if the equivalence relation is amenabl ...

Invariant means on topological semigroups

... means on the space C*(S) of all bounded continuous real-valued functions on S. Of course S is pseudocompact, then C(S) = C*(S) and the two problems are the same. Our main result is to the effect that, for a large class of semigroups S, every left invariant mean (if any exist) arises in a particularl ...

... means on the space C*(S) of all bounded continuous real-valued functions on S. Of course S is pseudocompact, then C(S) = C*(S) and the two problems are the same. Our main result is to the effect that, for a large class of semigroups S, every left invariant mean (if any exist) arises in a particularl ...

Abelian Categories

... • Note that in many concrete contexts, one would refer to the object K as the "kernel", rather than the morphism k. In those situations, K would be a subset of X, and that would be sufficient to reconstruct k as an inclusion map; in the nonconcrete case, in contrast, we need the morphism k to descri ...

... • Note that in many concrete contexts, one would refer to the object K as the "kernel", rather than the morphism k. In those situations, K would be a subset of X, and that would be sufficient to reconstruct k as an inclusion map; in the nonconcrete case, in contrast, we need the morphism k to descri ...

The local structure of compactified Jacobians

... singularities. Under the same assumption on d and g, the same authors computed the Kodaira dimension and the Itaka ﬁbration of J¯d,g ([6, Theorem 1.2]), and in [13], the present authors will extend that computation to all d, g. The authors also hope to use the results of this paper to study the sing ...

... singularities. Under the same assumption on d and g, the same authors computed the Kodaira dimension and the Itaka ﬁbration of J¯d,g ([6, Theorem 1.2]), and in [13], the present authors will extend that computation to all d, g. The authors also hope to use the results of this paper to study the sing ...

Algebra Qual Solutions September 12, 2009 UCLA ALGEBRA QUALIFYING EXAM Solutions

... Let z ∈ X\{x, y}. By double transitivity, there exists g ∈ G such that gx = y and gy = z. By the above paragraph, g ∈ H, so gh ∈ H. But ghx = z, so by the above paragraph applied to gh, anything mapping x to z is in H. Thus H = G. ¤ G.1. G7s1. Let G be a simple group containing an element of order 2 ...

... Let z ∈ X\{x, y}. By double transitivity, there exists g ∈ G such that gx = y and gy = z. By the above paragraph, g ∈ H, so gh ∈ H. But ghx = z, so by the above paragraph applied to gh, anything mapping x to z is in H. Thus H = G. ¤ G.1. G7s1. Let G be a simple group containing an element of order 2 ...

derived smooth manifolds

... Theorem 2.6). We did not include that as an axiom here, however, because it does not seem to us to be an inherently necessary aspect of a good intersection theory. The category of smooth manifolds does not have the general cup product formula because it does not satisfy Condition (2). Indeed, suppos ...

... Theorem 2.6). We did not include that as an axiom here, however, because it does not seem to us to be an inherently necessary aspect of a good intersection theory. The category of smooth manifolds does not have the general cup product formula because it does not satisfy Condition (2). Indeed, suppos ...

Representations of dynamical systems on Banach spaces

... systems and Banach spaces. A further indication for the success of this theory is the fact that it provides also a natural environment for the study of representations of topological groups and compact right topological semigroups on Banach spaces. Throughout our review there are some new results w ...

... systems and Banach spaces. A further indication for the success of this theory is the fact that it provides also a natural environment for the study of representations of topological groups and compact right topological semigroups on Banach spaces. Throughout our review there are some new results w ...

# Group action

In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes of the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.A group action is an extension to the notion of a symmetry group in which every element of the group ""acts"" like a bijective transformation (or ""symmetry"") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.If G is a group and X is a set, then a group action may be defined as a group homomorphism h from G to the symmetric group on X. The action assigns a permutation of X to each element of the group in such a way that the permutation of X assigned to the identity element of G is the identity transformation of X; a product gk of two elements of G is the composition of the permutations assigned to g and k.The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.