Group Theory and Its Application: Beamer Presentation (PPT)

Group Theory and Its Application : Beamer Presentation (PPT)

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.
2.
3.
4.
5.
6.
7.
8.
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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,
elds, modules, vector spaces, lattices, and
algebras.
UPC-A barcode
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4. GROUPS
Let G be a non-empty set andbe a binary operation dened on
it, then the structure (G,) is said to be a group, if the following
axioms are satised,
(i) Closure property :ab2G;8a;b2G
(ii) Associativity :The operationis associative on G. i.e.
a(bc) = (ab)c;8a;b;c2G
(iii) Existence of identity :There exists an unique element e2
G, such that
ae=a=ea;8a2G
e is called identity ofin G
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5. (iv) Existence of inverse :for each element a2G, there exist an
unique element b2G such that
ab=e=ba
The element b is called inverse of element a with respect toand
we writeb=a
1
Abelian Group
A group (G,) is said to be abelian or commutative, if
ab=ba8a;b2G
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6. Some Examples of Group
The set of all 33 matrices with real entries of the form
2
6
4
1a b
0 1c
0 0 1
3
7
5
is a group.
This group sometimes called the
prize-winning physicist Werner Heisenberg, is intimately related to
theHeisenberg uncertainity principleof quantum physics.
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7. Another example
The set of six transformationsf1;f2;f3;f4;f5;f6on the set of
complex numbers dened by
f1(z) =z;f2(z) =
1
z
;f3(z) = 1z;f4(z) =
z
z1
;
f5(z) =
1
1z
;f6(z) =
z1
z
:
forms a nite 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 nite group is called the order of a
group. It is denoted byo(G).
An innite group is a group of innite order.
e.g.,
1. G=f1;1g;thenGis an abelian group of order 2 with
respect to multiplication.
2. Zof integers is an innite group with respect to the
operation of addition butZis 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
LetGbe a group under multiplication. Let e be the identity
element inG. Suppose, a is any element ofG, then the least
positive integer n, if exist, such thata
n
=eis said to be order of
the elementa, 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 innite or zero order.
e.g.,
(i) The multiplicative groupG=f1;1;i;igof 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 groupZ=f:::;3;2;1;0;1;2;3; :::g
1:0 = 0)order of zero is one(nite).
butna6= 0 for any non zero integers a.
)o(a) is innite.
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11. Modular Arithmetic
Modular Arithmetic imports its concept from division algorithm
(a=qn+r;where 0r<n) and is an abstraction of method of
counting that we often use.
Modulo system
Letnbe a xed positive integer andaandbare two integers, we
deneab(modn);ifnj(ab) 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 xed
positive integers, then these are dened by
a+mb=r;0r<m;and
apb=r0r<p where r is the least non-negative
remainder ,wherna+banda:bdivided by m and p
respectively.
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12. Examples. f0;1;2;3; :::(n1)gof n elements is a nite
abelian group under addition modulo n.
Time-keeping on this clock uses arithmetic modulo 12.
(ii)Fermat's Little theorem :Ifpis prime, then
a
p1
1(modp) for 0<a<p:
(iii)Euler's theoremifaandnare co-prime, then
a
(n)
1(modn);
whereis 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 identication number has 12
digits. The rst six digits identify manufacturer, the
next ve digit identify the product, and the last is a
check.
An item with UPC identication
numbera1;a2;:::a12satises 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
Denition
A non-empty subsetHof a group (G;) is said to be subgroup
ofG, if (H;) is itself a group.
e.g., [f1,-1g, .] is a subgroup of [f1,-1,i,-ig.]
Criteria for a Subset to be a Subgroup
A non-empty subsetHof a groupGis a subgroup ofGif and
only if
(i)a;b2H)ab2H
(ii)a2H)a
1
2H;
wherea
1
is the inverse ofa2G
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15. Lagrange's Theorem
Statement
The order of each subgroup of a nite group is a divisor of the
order of the group.
i.e., LetHbe a subgroup of a nite groupGand let
o(G) =nando(H) =m;then
mjn (m divides n)
Since,f:H!aHandf:H!Hais one-one and onto.
)o(H) =o(H) =m
Now,G=H[Ha[Hb[Hc[:::;where a,b,c,...2G
) o(G) =o(H) +o(Ha) +o(Hb) +:::
)n=m+m+m+m+::::+ uptopterms (say)
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16. ) n=mp
)Order of the subgroup of a nite group is a divisor of the
order of the group.
> > >
?The converse of Lagrange's theorem is not true.
e:g:;
Consider the symmetric groupP4of permutation of degree 4.
Theno(P4) = 4! = 24 LetA4be the alternative group of even
permutation of degree 4. Then,o(A4) =
24
2
= 12. There exist no
subgroupHofA4, such thato(H) = 6, though 6 is the divisor of
12.
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17. Sylow's theorems
In the eld of nite group theory, theSylow theoremsare a
collection of theorems named after theNorwegian
mathematician Ludwig Sylow (1872)that give detailed
information about the number of subgroups of xed order that a
given nite group contains. The Sylow theorems form a
fundamental part of nite group theory and have very important
applications in the classication of nite simple groups.
Sylowp-subgroups
Leto(G) =p
m
n;wherepis the prime andm;nthe positive
integers such thatp-n:Then, a subgroupHofGis said to be a
sylowp-subgroup ofG;ifo(H) =p
m
ando(H) is the highest
power ofpthat divideso(G):
There are threeSylow theorem, and loosely speaking, they
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18. describe the following about a group'sp-subgroups:
1.Existence:In every group,p-subgroups of all possible sizes
exist.
2.Relationship:All maximalp-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 nite group such thatp
m
jo(G) andp
m+1
-o(G);
wherepis a prime number andmis a positive integer. Then,G
has subgroups of orderp;p
2
;p
3
; :::;p
m
.
e.g.,
Let a group of order 45fo(G) = 45gand since 45 = 3
2
5;then
Ghas 3-sylow subgroupHof order 9 and a 5-sylow subgroupKof
order 5.
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19. Sylow's Second Theorem
LetGbe a nite group andpbe a prime number such that
pjo(G). then, any two sylowp-subgroups ofGare conjugate.
Conjugate Subgroup: A subgroupHof a original groupG
has elementshi:Letxbe a xed element of the original group
Gwhich is not a member ofH. Then th transformation
xhix
1
(i= 1;2; :::) generates so called conjugate subgroup
xHx
1
:
Sylow's Third Theorem
The number of sylowp-subgroup inGfor a given primep, is
of the form 1 +kp, wherekis some non-negative integer and
(1 +kp)jo(G):
e.g., In above case,o(G) = 45 = 3
2
5
The number of sylow 3-subgroups is of the form 1 + 3ksuch
that 1 + 3kj3
2
5.
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20. Applications
1.
elds of mathematics and science.
2. algebraic topologyuses algebraic objects to
study topology.
3. Poincare conjectureasserts that the
fundamental groupof manifold, which encodes information
about connectedness, can be used to determine whether a
manifold is a sphere or not.
4.Algebraic number theorystudies various numberringsthat
generalize the set of integers.
5. Algebraic number theory, Andrew Wiles
provedFermat'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 unied form the largest possible
circle of formal relationships. In this eort 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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