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Abstract Algebra & its Applications.Abstract Algebra is the study of

 algebraic structures. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the parts of algebra. Solving of systems of linear equations, which led to linear algebra Linear algebra is the branch of mathematics concerning vector spaces and linear

mappings between such spaces.

•Solving of systems of linear equations, which led to linear algebra •Attempts to find formulae for solutions of general polynomial equations of higher degree that resulted in discovery of groups as abstract manifestations of symmetry •Arithmetical investigations of quadratic and higher degree forms that directly produced the notions of a ring and ideal.

Algebraic structures include

 groups,  rings  fields  modules,  vector spaces, lattices and 

algebra over a field

Algebraic structures

Binary operations are the keystone of algebraic structures studied in abstract algebra:

They are essential in the definitions of groups, monoids, semigroups, rings, and more.

A binary operation on a set S is a map which sends elements of the

 Cartesian product

S to S 

Binary operations

On the set M(2,2) of 2 × 2 matrices

with real entries, f(A, B) = A + B is a

binary operation since the sum of two

such matrices is another 2 × 2 matrix.

In abstract algebra, a magma (or groupoid) is a basic kind ofalgebraic structure.

Specifically, a magma consists of a set, M, equipped with a single binary operation,

 M × M → M. The binary operation must be closed by definition

but no other properties are imposed.

magma

Leonhard Euler --  algebraic operations on numbers--generalization of Fermat's little theorem  Friedric Gauss -  cyclic &general abelian groups

In 1870, Leopold Kronecker- abelian group-particularly, permutation groups.

 Heinrich M. Weber gave a similar definition that involved the cancellation property.

Lagrange resolvants by Lagrange. The remarkable Mathematicians

are ..Kronecker,Vandermonde,Galois,Augustin Cauchy ,

Cayley-1854-….Group may consists of Matrices.  

Early Group Theory

The end of the 19th and the beginning of the 20th century saw a tremendous shift in the methodology of mathematics.

Abstract algebra emerged around the start of the 20th century, under the name modern algebra.

Its study was part of the drive for more intellectual rigor in mathematics.

Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems.

MODERN ALGEBRA

Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra.

These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in Bartel van der Waerden's Moderne algebra.

The two-volume monograph published in 1930–1931 that forever changed for the mathematical world the meaning of the word…

“ algebra “ from the’ theory of equations’  to the ‘ theory of algebraic structures’.

Examples of algebraic structures with a single binary operation are:

Magmas

Quasigroups

Monoids

Semigroups

Groups

More complicated examples include: Rings Fields Modules Vector spaces Algebras over fields Associative algebras Lie algebras Lattices Boolean algebras

Because of its generality, abstract algebra is used in many fields of mathematics and science.

For instance, algebraic topology uses algebraic objects to study topologies.

The recently (As of 2006) proved Poincaré conjecture asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not.

 Algebraic number theory studies various number rings that generalize the set of integers.

Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem.

Applications

In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations.

In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system

The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system;

For instance, the number of force carriers in a theory is equal to dimension of the Lie algebra

And these bosons interact with the force they mediate if the Lie algebra is nonabelian.[2

Applications

Group-like structures Totality Associativity

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Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

 A representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication structures. The

The most prominent of these (and historically the first) is the representation theory of groups.

Representation theory

Let V be a vector space over a field F. The set of all invertible n × n matrices is a group

under matrix multiplication The representation theory of groups analyses a group

by describing ("representing") its elements in terms of invertible matrices.

This generalizes to any field F and any vector space V over F, with linear maps replacing matrices and compositionreplacing matrix multiplication:

There is a group GL(V,F) of automorphisms of V an associative algebra EndF(V) of all endomorphisms

of V, and a corresponding Lie algebra gl(V,F).

Definition

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