This document provides an overview of a seminar presentation on group theory and its applications. The presentation covers topics such as the definition of groups, order of groups and group elements, modular arithmetic, subgroups, Lagrange's theorem, and Sylow's theorems. It also discusses some examples of groups and applications of group theory in fields like algebraic topology, number theory, and physics. The presentation aims to introduce fundamental concepts in modern algebra through group theory.

1. Modern Algebra
Group theory and Its applications
M.Sc. Seminar Presentation
Course Code: MMS 13
By
Siraj Ahmad
M.Sc.(Mathematics)-Third Semester
Roll No. 1171080004
Department of Mathematics and Computer Science
School of Basic Sciences
Babu Banarasi Das University, Lucknow 226028, India
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2. Table of Contents
1. About Modern Algebra
2. Definition of Groups
3. Order of a group and order of an element
4. Modular Arithmetic
5. Subgroup
6. Lagrange’s theorem
7. Sylow’s Theorem
8. Applications
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3. Modern Algebra
Modern algebra, also called abstract
algebra, branch of mathematics concerned
with the general algebraic structure of
various sets (such as real numbers, complex
numbers, matrices, and vector spaces),
rather than rules and procedures for
manipulating their individual elements.
Algebraic structures include groups, rings,
fields, modules, vector spaces, lattices, and
algebras.
UPC-A barcode
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4. GROUPS
Let G be a non-empty set and ∗ be a binary operation defined on
it, then the structure (G,∗) is said to be a group, if the following
axioms are satisfied,
(i) Closure property :a ∗ b ∈ G, ∀ a, b ∈ G
(ii) Associativity :The operation ∗ is associative on G. i.e.
a ∗ (b ∗ c) = (a ∗ b) ∗ c, ∀ a, b, c ∈ G
(iii) Existence of identity : There exists an unique element e ∈
G, such that
a ∗ e = a = e ∗ a, ∀ a ∈ G
e is called identity of ∗ in G
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5. (iv) Existence of inverse : for each element a∈ G, there exist an
unique element b∈ G such that
a ∗ b = e = b ∗ a
The element b is called inverse of element a with respect to ∗ and
we write b = a−1
Abelian Group
A group (G,∗) is said to be abelian or commutative, if
a ∗ b = b ∗ a ∀ a, b ∈ G
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6. Some Examples of Group
The set of all 3×3 matrices with real entries of the form



1 a b
0 1 c
0 0 1



is a group.
This group sometimes called the Heisenberg group after the Nobel
prize-winning physicist Werner Heisenberg, is intimately related to
the Heisenberg uncertainity principle of quantum physics.
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7. Another example
The set of six transformations f1, f2, f3, f4, f5, f6 on the set of
complex numbers defined by
f1(z) = z, f2(z) =
1
z
, f3(z) = 1 − z, f4(z) =
z
z − 1
,
f5(z) =
1
1 − z
, f6(z) =
z − 1
z
.
forms a finite non-abelian group of order six with respect to the
composition known as the composition of the two functions or
product of two functions.
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8. Order of a Group and Order of an element of a group
Order of a Group
The number of element in a finite group is called the order of a
group. It is denoted by o(G).
An infinite group is a group of infinite order.
e.g.,
1. Let G = {1, −1}, then G is an abelian group of order 2 with
respect to multiplication.
2. The set Z of integers is an infinite group with respect to the
operation of addition but Z is not a group with respect to
multiplication.
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9. Order of an element of a group
Order of an Element of a Group
Let G be a group under multiplication. Let e be the identity
element in G. Suppose, a is any element of G, then the least
positive integer n, if exist, such that an = e is said to be order of
the element a, which is represented by
o(a) = n
In case, such a positive integer n does not exist, we say that the
element a is of infinite or zero order.
e.g.,
(i) The multiplicative group G = {1, −1, i, −i} of fourth roots of
unity, have order of its elements
(1)1
= 1 ⇒ o(1) = 1 9/21

10. (−1)2
= 1 ⇒ o(−1) = 2
(i)4
= 1 ⇒ o(i) = 4
(−i)4
= 1 ⇒ o(−i) = 4
respectively.
(ii) The additive group Z = {..., −3, −2, −1, 0, 1, 2, 3, ...}
1.0 = 0 ⇒ order of zero is one(finite).
but na = 0 for any non zero integers a.
⇒ o(a) is infinite.
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11. Modular Arithmetic
Modular Arithmetic imports its concept from division algorithm
(a = qn + r, where 0 ≤ r < n) and is an abstraction of method of
counting that we often use.
Modulo system
Let n be a fixed positive integer and a and b are two integers, we
define a ≡ b(modn), if n | (a − b) and read as, ”a is congruent to
b mod n”.
Addition modulo m and Multiplication modulo p
Let a and b are any two integers and m and p are fixed
positive integers, then these are defined by
a +m b = r, 0 ≤ r < m, and
a ×p b = r 0 ≤ r < p where r is the least non-negative
remainder ,whern a + b and a.b divided by m and p 11/21

12. Examples. (i) The set {0, 1, 2, 3, ...(n − 1)} of n elements is a finite
abelian group under addition modulo n.
Time-keeping on this clock uses arithmetic modulo 12.
(ii) Fermat’s Little theorem : If p is prime, then
ap−1 ≡ 1(modp) for 0 < a < p.
(iii) Euler’s theorem if a and n are co-prime, then
aφ(n)
≡ 1(modn),
where φ is Euler’s totient function.
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13. Another Application of Modular Arithmetic
Barcode, also known as Universal Product Code
(UPC). A UPC-A identification number has 12
digits. The first six digits identify manufacturer, the
next five digit identify the product, and the last is a
check.
An item with UPC identification
number a1, a2, ...a12 satisfies the
condition
(a1, ...a12)(3, 1, ...3, 1)mod10 = 0
Now suppose a single error is made in entering the
number in computer, it won’t satisfy the condition. 13/21

14. Subgroup
Definition
A non-empty subset H of a group (G, ∗) is said to be subgroup
of G, if (H, ∗) is itself a group.
e.g., [{1,-1}, .] is a subgroup of [{1,-1,i,-i} .]
Criteria for a Subset to be a Subgroup
A non-empty subset H of a group G is a subgroup of G if and
only if
(i) a, b ∈ H ⇒ ab ∈ H
(ii) a ∈ H ⇒ a−1
∈ H,
where a−1
is the inverse of a ∈ G
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15. Lagrange’s Theorem
Statement
The order of each subgroup of a finite group is a divisor of the
order of the group.
i.e., Let H be a subgroup of a finite group G and let
o(G) = n and o(H) = m, then
m | n (m divides n)
Since, f : H → aH and f : H → Ha is one-one and onto.
⇒ o(H) = o(H) = m
Now, G = H ∪ Ha ∪ Hb ∪ Hc ∪ ..., where a,b,c,...∈ G
⇒ o(G) = o(H) + o(Ha) + o(Hb) + ...
⇒ n = m + m + m + m + ....+ upto p terms (say)
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16. ⇒ n = mp
⇒ Order of the subgroup of a finite group is a divisor of the
order of the group.
The converse of Lagrange’s theorem is not true.
e.g.,
Consider the symmetric group P4 of permutation of degree 4.
Then o(P4) = 4! = 24 Let A4 be the alternative group of even
permutation of degree 4. Then, o(A4) =
24
2
= 12. There exist no
subgroup H of A4, such that o(H) = 6, though 6 is the divisor of
12.
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17. Sylow’s theorems
In the field of finite group theory, the Sylow theorems are a
collection of theorems named after the Norwegian
mathematician Ludwig Sylow (1872) that give detailed
information about the number of subgroups of fixed order that a
given finite group contains. The Sylow theorems form a
fundamental part of finite group theory and have very important
applications in the classification of finite simple groups.
Sylow p-subgroups
Let o(G) = pmn, where p is the prime and m, n the positive
integers such that p n. Then, a subgroup H of G is said to be a
sylow p-subgroup of G, if o(H) = pm and o(H) is the highest
power of p that divides o(G).
There are three Sylow theorem, and loosely speaking, they 17/21

18. describe the following about a group’s p-subgroups:
1. Existence: In every group, p-subgroups of all possible sizes
exist.
2. Relationship: All maximal p-subgroups are conjugate.
3. Number: There are strong restriction on the number of
p-subgroups a group can have.
Sylow’s First Theorem
Let G be a finite group such that pm | o(G) and pm+1 o(G),
where p is a prime number and m is a positive integer. Then, G
has subgroups of order p, p2, p3, ..., pm.
e.g.,
Let a group of order 45 {o(G) = 45} and since 45 = 32 × 5, then
G has 3-sylow subgroup H of order 9 and a 5-sylow subgroup K of
order 5.
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19. Sylow’s Second Theorem
Let G be a finite group and p be a prime number such that
p | o(G). then, any two sylow p-subgroups of G are conjugate.
Conjugate Subgroup: A subgroup H of a original group G
has elements hi . Let x be a fixed element of the original group
G which is not a member of H. Then th transformation
xhi x−1
(i = 1, 2, ...) generates so called conjugate subgroup
xHx−1
.
Sylow’s Third Theorem
The number of sylow p-subgroup in G for a given prime p, is
of the form 1 + kp, where k is some non-negative integer and
(1 + kp) | o(G).
e.g., In above case, o(G) = 45 = 32
× 5
The number of sylow 3-subgroups is of the form 1 + 3k such
that 1 + 3k | 32
× 5.
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20. Applications
1. Because of its generality, abstract algebra is used in many
fields of mathematics and science.
2. For instance,algebraic topology uses algebraic objects to
study topology.
3. The recently proved Poincare conjecture asserts that the
fundamental group of manifold, which encodes information
about connectedness, can be used to determine whether a
manifold is a sphere or not.
4. Algebraic number theory studies various number rings that
generalize the set of integers.
5. Using tools of Algebraic number theory, Andrew Wiles
proved Fermat’s Last Theorem.
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21. Professor Einstein Writes in Appreciation of a
Fellow-Mathematician.
Pure mathematics is, in its way, the poetry of logical ideas.
One seeks the most general ideas of operation which will bring
together in simple, logical and unified form the largest possible
circle of formal relationships. In this effort toward logical
beauty spiritual formulas are discovered necessary for the
deeper penetration into the laws of nature.
ALBERT EINSTEIN.
Princeton University, May 1, 1935.
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