Abstract algebra is the study of algebraic structures like groups, rings, and fields. It emerged in the early 20th century to make algebra more rigorous and abstract. Key developments included the study of symmetry in polynomial equations, which led to the concepts of groups, and the algebraic investigation of quadratic and higher-degree equations, which produced the ideas of rings and ideals. Now abstract algebra is used throughout mathematics and in fields like physics, where group theory can simplify differential equations and describe system symmetries.

1. WELCOME

2. 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.

3. •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.

4. Algebraic structures
include
 groups,
 rings
 fields
 modules,
 vector spaces, lattices and algebra over a
field

5.  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

6.  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.

7.  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.

8.  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.

9.  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.

10.  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’.

11.  Examples of algebraic structures with a
single binary operation are:
 Magmas
 Quasigroups
 Monoids
 Semigroups
 Groups

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

13.  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.

14.  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

15.  Group-like structures Totality Associativity
Identity Divisibility Commutativity Semicategory Unneeded Required
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Required ^α Closure, which is used in many sources, is an equivalent
axiom to totality, though defined differently

16. Group-like structures
Totalityα Associativity Identity Divisibility
Commutativit
y
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17.  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.

18.  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).

19. THANK YOU