Algebra part - Georgia Tech Math
... of order 5. Solution. (a) Let G act on X = G by conjugation. The size of the conjugacy class of g ∈ G is |G|/|C(g)| where C(g) is the centralizer of g in G. Thus, the size of the conjugacy class is 5k for k = 1, 2, 3. k = 1 iff g ∈ Z and |Z| divides 125. Since 125 = |X| and 5 divides the size of eve ...
... of order 5. Solution. (a) Let G act on X = G by conjugation. The size of the conjugacy class of g ∈ G is |G|/|C(g)| where C(g) is the centralizer of g in G. Thus, the size of the conjugacy class is 5k for k = 1, 2, 3. k = 1 iff g ∈ Z and |Z| divides 125. Since 125 = |X| and 5 divides the size of eve ...
On Top Spaces
... Definition 4.1 If T , and S are two top spaces, then a homomorphism f : T → S is called a morphism if it is also a C ∞ map. Definition 4.2 A top generalized subgroup N of a top space T is called a top generalized normal subgroup of T if there exists a top space S and a morphism f : T → S such that, ...
... Definition 4.1 If T , and S are two top spaces, then a homomorphism f : T → S is called a morphism if it is also a C ∞ map. Definition 4.2 A top generalized subgroup N of a top space T is called a top generalized normal subgroup of T if there exists a top space S and a morphism f : T → S such that, ...
Chapter 12 Algebraic numbers and algebraic integers
... For example, 2 and i/2 are algebraic. A complex number is said to be transcendental if it is not algebraic. Both e and π are transcendental. It is in general extremely difficult to prove a number transcendental, and there are many open problems in this area, eg it is not known if π e is transcendent ...
... For example, 2 and i/2 are algebraic. A complex number is said to be transcendental if it is not algebraic. Both e and π are transcendental. It is in general extremely difficult to prove a number transcendental, and there are many open problems in this area, eg it is not known if π e is transcendent ...
(pdf)
... addition. It has infinite order, and the identity element e is the number 0. However, Z is not a group under multiplication. The identity in this case would be 1, but most integers do not have an inverse in Z. For example, 21 , the multiplicative inverse of 2, is not an integer. Note that, when the ...
... addition. It has infinite order, and the identity element e is the number 0. However, Z is not a group under multiplication. The identity in this case would be 1, but most integers do not have an inverse in Z. For example, 21 , the multiplicative inverse of 2, is not an integer. Note that, when the ...
Trivial remarks about tori.
... X∗ (T ) = T (F )/T (O), where O is the integers of F , the map being the following: given φ ∈ X∗ (T ), φ is a map GL1 → T , and one evaluates it at a uniformiser; the resulting element of T (F )/T (O) is well-defined. As a consequence we have X ∗ (Tb) = T (F )/T (O). ...
... X∗ (T ) = T (F )/T (O), where O is the integers of F , the map being the following: given φ ∈ X∗ (T ), φ is a map GL1 → T , and one evaluates it at a uniformiser; the resulting element of T (F )/T (O) is well-defined. As a consequence we have X ∗ (Tb) = T (F )/T (O). ...
Posets 1 What is a poset?
... cannot be proved from the Zermelo–Fraenkel axioms alone (assuming their consistency), but it is strictly weaker than the Axiom of Choice, that is, the Axiom of Choice (or Zorn’s Lemma) cannot be proved from the Zermelo–Fraenkel axioms and this assumption. In other words, assuming the axioms consiste ...
... cannot be proved from the Zermelo–Fraenkel axioms alone (assuming their consistency), but it is strictly weaker than the Axiom of Choice, that is, the Axiom of Choice (or Zorn’s Lemma) cannot be proved from the Zermelo–Fraenkel axioms and this assumption. In other words, assuming the axioms consiste ...
Algebra in Coding
... 7. Let x, y be non-zero elements of GF(16) (α is primitive satisfying α4 +α+1 = 0). Given x+y = α14 and x3 + y 3 = α, find x and y. 8. Consider GF(16) with α primitive satisfying α4 + α + 1 = 0. (a) Find all solutions to the simultaneous equations x + y = α3 and x2 + y 2 = α6 . (b) Find all solution ...
... 7. Let x, y be non-zero elements of GF(16) (α is primitive satisfying α4 +α+1 = 0). Given x+y = α14 and x3 + y 3 = α, find x and y. 8. Consider GF(16) with α primitive satisfying α4 + α + 1 = 0. (a) Find all solutions to the simultaneous equations x + y = α3 and x2 + y 2 = α6 . (b) Find all solution ...
4 Modular Arithmetic
... Public key cryptography. Today, we use more powerful encryption methods that give a more flexible way to transmit secret information. We call this public key cryptography which roughly works as follows. As before, we have a sender, called Alice, and a receiver, called Bob. Both Alice and Bob have a ...
... Public key cryptography. Today, we use more powerful encryption methods that give a more flexible way to transmit secret information. We call this public key cryptography which roughly works as follows. As before, we have a sender, called Alice, and a receiver, called Bob. Both Alice and Bob have a ...
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation; the addition of any two integers forms another integer. The abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A theory has been developed for finite groups, which culminated with the classification of finite simple groups announced in 1983. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.