WHY GROUPS? Group theory is the study of symmetry. When an
... Group theory is the study of symmetry. When an object appears symmetric, group theory can help us study it. We apply the label “symmetric” to anything that stays invariant under some transformations. This could apply to geometric figures (a circle is highly symmetric, being invariant under any rotat ...
... Group theory is the study of symmetry. When an object appears symmetric, group theory can help us study it. We apply the label “symmetric” to anything that stays invariant under some transformations. This could apply to geometric figures (a circle is highly symmetric, being invariant under any rotat ...
Finite Fields - (AKA Galois Fields)
... an irreducible polynomial p(x) ∈ Fp [x], and define congruence in Fp [x] by declaring that a(x) ≡ b(x) (mod p(x)) if and only if a(x) − b(x) is a multiple of p(x). Let Fq be the set of residues mod p(x). This is just the set of all polynomials of degree strictly less than r . It is easy to see that ...
... an irreducible polynomial p(x) ∈ Fp [x], and define congruence in Fp [x] by declaring that a(x) ≡ b(x) (mod p(x)) if and only if a(x) − b(x) is a multiple of p(x). Let Fq be the set of residues mod p(x). This is just the set of all polynomials of degree strictly less than r . It is easy to see that ...
Algebra Qualifying Exam January 2015
... 5. Let u = 4 2, and let D4 be the dihedral group of rigid motions of a square. Recall that D4 can be described by generators and relations as follows: D4 =< x, y | x4 = e, y 2 = e, yx = x3 y > where x, y denote the generators of the group, and e denotes the identity element of the group. a. Prove th ...
... 5. Let u = 4 2, and let D4 be the dihedral group of rigid motions of a square. Recall that D4 can be described by generators and relations as follows: D4 =< x, y | x4 = e, y 2 = e, yx = x3 y > where x, y denote the generators of the group, and e denotes the identity element of the group. a. Prove th ...
Math 110 Homework 9 Solutions
... pick a polynomial q(x) ∈ Fp [x] representing it. Note that q(x) and P (x) are relatively prime. Then as with Euclidean algorithm in Z, we can find solutions r(x) and t(x) such that P (x)r(x) + q(x)t(x) = 1. Looking at this modulo P (x), we have a multiplicative inverese for q(x). (e) Consider x3 and ...
... pick a polynomial q(x) ∈ Fp [x] representing it. Note that q(x) and P (x) are relatively prime. Then as with Euclidean algorithm in Z, we can find solutions r(x) and t(x) such that P (x)r(x) + q(x)t(x) = 1. Looking at this modulo P (x), we have a multiplicative inverese for q(x). (e) Consider x3 and ...
Examples, Binary Structures, Isomorphisms
... There is no universal way to prove that two binary algebraic structures (S, ∗) and (S 0 , ∗0 ) are not isomorphic. However, if S and S 0 have different cardinalities, then surely there is no one-to-one and onto map between S and S 0 , and thus S and S 0 are not isomorphic. Ex 1.24. (Zn , +n ) and (Z ...
... There is no universal way to prove that two binary algebraic structures (S, ∗) and (S 0 , ∗0 ) are not isomorphic. However, if S and S 0 have different cardinalities, then surely there is no one-to-one and onto map between S and S 0 , and thus S and S 0 are not isomorphic. Ex 1.24. (Zn , +n ) and (Z ...
Modules Over Principal Ideal Domains
... analogous definitions from vector spaces. The first part of the paper will be full of definition, but once in place, we will have a language to generalize our important results from group theory. First though, we must address what a module is. A module can be thought of as a generalization of a vect ...
... analogous definitions from vector spaces. The first part of the paper will be full of definition, but once in place, we will have a language to generalize our important results from group theory. First though, we must address what a module is. A module can be thought of as a generalization of a vect ...
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.