1. Chapter 7 Algebraic Structures
Discrete Mathematics: A Concept-based Approach

2. History of Group Theory
The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than four. The 19th-century French mathematician Evariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry groups of its roots (also called solutions). Geometry is a field in which groups are used systematically, especially symmetry groups.
Discrete Mathematics: A Concept-based Approach

3. In mathematics, a group is viewed as consisting of a set of elements together with an operation to combine any two of its elements to get new element that also belongs to the same set while satisfying four conditions, namely, group axioms, closure, associativity, identity and invertibility. The group consists of the set of transformations that leaves the object unchanged in spite of combination of transformations and performing them in a sequence. The Lie groups belong to symmetry groups, which find application in the study of standard model of particle physics. The Point groups are used to help understand symmetry phenomena in molecular chemistry.
Discrete Mathematics: A Concept-based Approach

4. Main classes of groups : The groups are divided into
five main classes, listed below.
Permutation groups
Matrix groups
Transformation groups
Abstract groups
Topological and algebraic groups
Discrete Mathematics: A Concept-based Approach

5. Binary Operations and General Properties
Let S- be a non-empty set and * (read as star) be an operation on S. The operation on the set is a rule, which assigns to each ordered pair of elements of the set, a unique element of S.
Closure Property
Consider a binary operation, . The operation * is said to
be closed, if for all
The new element also belongs to S.
Discrete Mathematics: A Concept-based Approach

6. Example : A set of integers Z is closed with respect to the binary operations, namely, addition, multiplication and subtraction but not with respect to division. Example: The set of odd integers is not closed with respect to addition, since sum of two odd integers is an even, which is not the member of the set.
Discrete Mathematics: A Concept-based Approach

7. Associative Property Consider a binary operation
Associative Property Consider a binary operation *. For any Example The addition (+) and multiplication (.) are Associative in the following sets . N = The set of natural numbers, I or Z = The set of Integers, Q = The set of Rational, R = The set of real, C = The set of Complex numbers. (a+b) + c = a+ (b+c) , (a.b) . c = a. (b.c)
Discrete Mathematics: A Concept-based Approach

8. Existence of Identity Element
Consider an element e , such that. Then the element
is called the identity element of S with respect to the
binary operation . For example , 0 and 1 are the identity
elements of Z with respect to the operations of addition
and multiplication respectively.
Existence of Inverse:
Consider an element The element , is called
the inverse of a under the operation * .such that
Discrete Mathematics: A Concept-based Approach

9. Let G – be a non-empty set and be a binary operation on G.
Groups
Let G – be a non-empty set and be a binary operation on G.
Then G- is called a group under the operation if the following
properties hold good.
Closure Property:
Associative Property:
Existence of Identity element:
Existence of Inverse element:
In addition to the foresaid four properties, if the group satisfies the commutative property, then the group is called commutative group or abelian group.
Discrete Mathematics: A Concept-based Approach

10. Example : Show that the four matrices forms a group under matrix multiplication. Now G = {I,A,B,C} Proof: IA = AI= A, IB =BI= B, IC= CI = C. I is Identity matrix. Closure Property holds good. It can be shown that BA=C, AC=B, BC = A, AA=I, BB=I, CC=I by virtue of matrix multiplications. The composition operation is given in the following group table.
Discrete Mathematics: A Concept-based Approach

11. Composition operation.B C
Discrete Mathematics: A Concept-based Approach

12. Identity axiom: I is the identity element.
The entries in the table are the same as the elements of set G. The closure property is satisfied.
Associative axiom. A(BC) = A(A) = I, (AB)C = C(C) = I. Associative property is satisfied.
Identity axiom: I is the identity element.
Inverse axiom: Every element of G has an inverse.
Inverses of I, A, B, C are respectively I, A, B, C.
Hence G – is a group under multiplication
Discrete Mathematics: A Concept-based Approach

13. Example : In a group G having more than one element
then Prove that G- is abelian.
Proof: Take any , then from hypothesis,
( ab)2 = ab.
Now a(ab)b= (aa) (bb) = a2 b2
= ab = (ab)2= (ab)(ab) = a(ba)b
or a(ab)b= a(ba)b.
The left and right cancellation property holds good.
ab= ba , this shows that G- is abelian.
Discrete Mathematics: A Concept-based Approach

14. Semi-groups and monoids Let S- be a nonempty set and is a binary operation on S, then the algebraic system (S,*) is called a semi-group , if the operation * is associative. The algebraic system is called semigroup. It is to note that since the characteristic property of a binary operation on a set S is the closure property, it is not necessary to mention it explicitly when algebraic system is defined.
Discrete Mathematics: A Concept-based Approach

15. Monoid:If a semi- group (M,
Monoid:If a semi- group (M,*) has an identity element with respect to the operation * , then (M,*) is called a monoid. The algebraic system is called a monoid. The set of positive even numbers is a semi-group with respect to the binary operation addition and Multiplication.
Discrete Mathematics: A Concept-based Approach

16. 2-(5 – 6) = 2 - (-1) = 3, which is not the element of Z .
Example: The set of negative integers is not a semi-group. Hence, the operation - is closed in Z. But the operation is not associative.
Consider a = 2 , b = 5, and c = 6
(2-5) - 6 = -3 – 6 = - 9
2-(5 – 6) = 2 - (-1) = 3, which is not the element of Z .
Example : The set of natural numbers N ={ 1,2,3,----} is a semi-group under the operation addition, but not a monoid, since the identity element does not exists i.e., .
Discrete Mathematics: A Concept-based Approach

17. The set S3 of all permutations of finite set containing
Symmetric Group
The set S3 of all permutations of finite set containing
three elements forms a non-abelian group under
composition of permutations. Where S = { 1, 2, 3 }. The
possible permutations are 3! = 6.
Cayley’s composition table of permutations on S3 is given
in Table.
Discrete Mathematics: A Concept-based Approach

18. Cayley’s composition table of permutations on S3* P1 P2 P3 P4 P5 P6
Discrete Mathematics: A Concept-based Approach

19. Geometrical Meanings of six permutations, S = { 1, 2, 3}
The geometric meaning of six permutations, namely,
dihedral group is given as under.
Dihedral Group:
The set of transformations due to all rigid motions of a
regular polygon of n-sides resulting in identical polygons but
with different vertex names under the binary operation of
right composition * is a group and is called dihedral group.
Let us consider a three sided regular polygon i.e., an
equilateral triangle, whose vertices are named as 1, 2, 3.
Discrete Mathematics: A Concept-based Approach

20. Rotations of a triangle: The rotation of Equilateral triangle about its centroid through angles in the anticlockwise direction and reflection of the triangle about its lines of symmetry is made. ( The process is known as rigid motion).
Discrete Mathematics: A Concept-based Approach

21. Example: Let G = { 1, i , -1 , -i } then (G , * ) is a cyclic group.
Cyclic Groups:
A group (G , * ) is said to cyclic group if there exists an
element , such that every element of G , can be expressed as
some integral power of a. Every element x of G can be
expressed as for some integer n. The cyclic group is said to
be generated by a or a is generator of G, we write G = {a}.
Example: Let G = { 1, i , -1 , -i } then (G , * ) is a cyclic group.
The generator is i.
Similarly, -i, is also a generator , and G = { -i}.
Discrete Mathematics: A Concept-based Approach

22. Homomorphisms Homomorphism is derived from the Greek word “ homoiosorphe”, (meaning similar form), is defined as a special correspondence between the members (elements) of two algebraic systems, such as two groups, two rings, or two fields. Two homomorphic systems have the same basic structure, and, while their elements and operations may appear entirely different. Results on one system often apply as well to the other system. Thus, if a new system can be shown to be homomorphic to a known system, certain known features of one can be applied to the other, Thereby simplifying the analysis of the new system.
Discrete Mathematics: A Concept-based Approach

23. Definition : Let (G,. ) and(H,
Definition : Let (G,*) and(H,*) be the group and its subgroup then the function f : G H is a homomorphism and the following relation holds, . A homomorphism is an isomorphism if it is bijective equivalently, if it has an inverse. If G and H are groups, G and H are isomorphic if there is an isomorphism. f : G H . Isomorphic groups are the same as groups.
Discrete Mathematics: A Concept-based Approach

24. Groups G and H are not isomorphic if they have different orders, or if one satisfies the group property that the other doesn't. For example, two groups are not isomorphic if one is abelian and the other is not; two groups are not isomorphic if the orders of elements of one are not the same as the orders of elements of the other.
Discrete Mathematics: A Concept-based Approach

25. Example: Consider logs and exponentials,
, is a homomorphism from the real
set under addition to the positive real set under
multiplication. The operation on the domain R is
addition, while the operation on the range is
multiplication. Therefore, we can show that exp is a
homomorphism,
But and ,
Discrete Mathematics: A Concept-based Approach

26. hence from identity exp is a homomorphism
hence from identity exp is a homomorphism. Applications of group theory Application of group theory is widespread. Galois theory uses groups to describe the symmetries of the roots of a polynomial. The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the corresponding Galois group. For example, S5, the symmetric group in 5 elements, is not solvable, which implies that the general equation of degree (n>5 ) cannot be solved by radicals as done in case lower orders. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory.
Discrete Mathematics: A Concept-based Approach

27. The concept of the Lie group, named after mathematician Sophus Lie, is important in the study of differential equations and manifolds. These describe the symmetries of continuous geometric and analytical structures. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noethers’s theorem, every symmetry of a physical system corresponds to a conservation law of the system.
Discrete Mathematics: A Concept-based Approach

28. Summary
Modern group theory is a research area in mathematics. Groups are algebraic systems that find applications in coding theory. Cryptography is an active research area in the Internet era, where secured transmission of information is a challenge.
Haar measures, that is integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques. In material science, groups are used to classify crystal structures, regular polyhedral, and the symmetry of molecules.
Discrete Mathematics: A Concept-based Approach