Math 3121 Abstract Algebra I
... product of an even number of transpositions, and odd if it can be expressed as a product of an odd number of permutations. Theorem: The parity of a permutation is even or odd, but not both. Proof: We show thatFor any positive integer n, parity is a homomorphism from Sn to the group ℤ2, where 0 repre ...
... product of an even number of transpositions, and odd if it can be expressed as a product of an odd number of permutations. Theorem: The parity of a permutation is even or odd, but not both. Proof: We show thatFor any positive integer n, parity is a homomorphism from Sn to the group ℤ2, where 0 repre ...
November 3
... a function on the domain R \ {0}. But if we extended it to a function of all of R, say by setting f (0) = 0, then it would not be locally bounded at 0. That is, it is not bounded in any neighborhood of 0. We can never extend f (x) = 1/x to a function continuous at zero because this would contradict ...
... a function on the domain R \ {0}. But if we extended it to a function of all of R, say by setting f (0) = 0, then it would not be locally bounded at 0. That is, it is not bounded in any neighborhood of 0. We can never extend f (x) = 1/x to a function continuous at zero because this would contradict ...
10. The isomorphism theorems We have already seen that given
... Proof. Consider the natural map G −→ G/H. The kernel, H, contains K. Thus, by the universal property of G/K, it follows that there is a homomorphism G/K −→ G/H. This map is clearly surjective. In fact, it sends the left coset gK to the left coset gH. Now suppose that gK is in the kernel. Then the le ...
... Proof. Consider the natural map G −→ G/H. The kernel, H, contains K. Thus, by the universal property of G/K, it follows that there is a homomorphism G/K −→ G/H. This map is clearly surjective. In fact, it sends the left coset gK to the left coset gH. Now suppose that gK is in the kernel. Then the le ...
Normal Subgroups The following definition applies. Definition B.2: A
... a homomorphism since f (m + n) = am+n = am· an= f (m) · f (n) The image of f is gp(a), the cyclic subgroup generated by a. By Theorem B.9, gp(a) ∼= Z/K where K is the kernel of f. If K = {0}, then gp(a) = Z. On the other hand, if m is the order of a, then K ={multiples of m}, and so gp(a) ∼= Zm. In ...
... a homomorphism since f (m + n) = am+n = am· an= f (m) · f (n) The image of f is gp(a), the cyclic subgroup generated by a. By Theorem B.9, gp(a) ∼= Z/K where K is the kernel of f. If K = {0}, then gp(a) = Z. On the other hand, if m is the order of a, then K ={multiples of m}, and so gp(a) ∼= Zm. In ...
1 An introduction to homotopy theory
... defines a homotopy of paths γ0 ⇒ γ1 . Hence there is a single homotopy class of paths joining p, q, and so Π1 (X) maps homeomorphically via the source and target maps (s, t) to X × X, and the groupoid law is (x, y) ◦ (y, z) = (x, z). This is called the pair groupoid over X. The fundamental group π1 ...
... defines a homotopy of paths γ0 ⇒ γ1 . Hence there is a single homotopy class of paths joining p, q, and so Π1 (X) maps homeomorphically via the source and target maps (s, t) to X × X, and the groupoid law is (x, y) ◦ (y, z) = (x, z). This is called the pair groupoid over X. The fundamental group π1 ...
math.uni-bielefeld.de
... by consideration of the correspondence given by the closure in X × X of the graph of a given rational map X → X. Remark 1.3. Assume that d(X) = 2n . Although we have Theorem 1.2, we do not know whether the variety X is 2-incompressible in the sense of [7, §7]. Note that the only known proof of p-inc ...
... by consideration of the correspondence given by the closure in X × X of the graph of a given rational map X → X. Remark 1.3. Assume that d(X) = 2n . Although we have Theorem 1.2, we do not know whether the variety X is 2-incompressible in the sense of [7, §7]. Note that the only known proof of p-inc ...
GLOBALIZING LOCALLY COMPACT LOCAL GROUPS 1
... the totally disconnected case more is true (but the result is much less deep): Proposition 1.2. If G is locally compact and totally disconnected, then some restriction of G is a compact topological group. This is proved just as in the global case for which we refer to [7], p. 54. This proposition wi ...
... the totally disconnected case more is true (but the result is much less deep): Proposition 1.2. If G is locally compact and totally disconnected, then some restriction of G is a compact topological group. This is proved just as in the global case for which we refer to [7], p. 54. This proposition wi ...
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.