Abstract Algebra

1. WELCOME

2. MEGA - 2015
(Mathematical Excellence Gears Advancement-2015)
SRI SARADA NIKETAN COLLEGE FOR WOMEN
Amaravathipudur, Karaikudi -630301 .
DEPARTMENT OF MATHEMATICS
State Level Workshop
‘Abstract Algebra and its Applications’
28th August , 2015.

3. Presentation on
‘Abstract Algebra and its Applications’
Presented by
Dr.S.SelvaRani, Principal
Sri Sarada Niketan College For Women
Amaravathipudur
Venue : Nivedita Hall
Sri Sarada Niketan College for Women,
Date : 28th August , 2015

4. Abstract Algebra is the study of
.
 The term abstract algebra was coined in the early
20th century to distinguish this area of study from
the the parts of algebra.
 Solving of systems of linear equations, which led
to
 Linear algebra is the branch
of concerning and
between such spaces.

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

6. 

7. on numbers-
- of theorem
Friedric Gauss - &general groups
 In 1870, - abelian group-
particularly, permutation groups.
 gave a similar definition that
involved the .
 Lagrange resolvants by Lagrange.
 The remarkable Mathematicians are
..Kronecker,Vandermonde,Galois,Augustin Cauchy ,
Cayley-1854-….Group may consists of Matrices.

8.  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 in mathematics.
 Initially, the assumptions in classical , on
which the whole of mathematics (and major parts
of the ) depend, took the form
of .

9. and , who had
considered ideals in commutative rings, and
of and , 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
's Moderne algebra.
 The two-volume mo 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’.

10.  Examples of algebraic structures with a
single are:

11.  More complicated examples include:

12.  Binary operations are the keystone of algebraic
structures studied in
 A binary operation is an operation that applies to two
quantities or expressions and .
 A binary operation on a is a map such
that
1. is defined for every pair of elements in , and
2. uniquely associates each pair of elements in to
some element of .

13.  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.

14.  In , a magma (or groupoid) is a basic
kind of .
 Specifically, a magma consists of a , M, equipped
with a single ,
 M × M → M.
 The binary operation must be by definition
but no other properties are imposed.

15. Group-like structures
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16.  Representation theory is a branch
of that studies
by representing their as
of , and
studies over these abstract algebraic
structures.
 A representation makes an abstract algebraic
object more concrete by describing its elements
by and the in terms
of and
structures.
 The most prominent of these (and historically
the first) is the

17.  Let V be a over a F.
 The set of all n × n matrices is a group under
 The 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 replacing matrices and
matrix multiplication:
 There is a group of of V
 an associative algebra EndF(V) of all endomorphisms of V, and a
corresponding Lie algebra gl(V,F).

18. Representation theory studies symmetry in Linear spaces.
• It has many applications, ranging from number theory to
geometry, probability theory, quantum mechanics and quantum
field theory.
•Representation theory was born in 1896 in the work of the German
mathematician F. G. Frobenius.
•And major contributors are : Dedekind, Burnside and A.H.Clifford.
Applications & Contributors

19.  Because of its generality, abstract algebra is used in
many fields of mathematics and science.
 For instance, uses algebraic
objects to study topologies.
 The recently (As of 2006) proved
asserts that the of a
manifold, which encodes information about
connectedness, can be used to determine whether a
manifold is a sphere or not.
 studies various
number that generalize the set of integers.
 Using tools of
proved .

20.  In physics, groups are used to represent symmetry
operations, and the usage of group theory could
simplify differential equations.
 In , the requirement of
can be used to deduce the equations
describing a system
 The groups that describe those symmetries are
, and the study of Lie groups and Lie
algebras reveals much about the physical system;
 For instance, the number of in a
theory is equal to dimension of the Lie algebra
 And these interact with the force they
mediate if the Lie algebra is nonabelian.[2

21. THANK YOU