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. Presentation on Introduction to Groups Dr. K. SUVARNA Lecturer in Mathematics III Sem, Abstract Algebra D.K. Govt. college for women(A) Nellore. Ph : 9985039953 .

The study of Algebraic Structures which have been subjected to Axiomatic development is called “Abstract Algebra or Group Theory”. Basic ingredients of Group Theory are Sets, Relations and Mappings. A set with a method of combination of elements of it called an Algebraic structure.

‘+’ is the binary operation in N. Let s be a non-empty set If f : sxs →s is a mapping then f is called “Binary operation” in S or “Binary composition” in S. i.e., (a , b) є SXS => f(a , b) є S i.e., a є S , b є S => a.b є S then ‘.’ is called a “Binary operation” in the set S. which is known as “Closure law”. Ex: Let N be the set of Natural numbers and for any a є N , b є N if a + b є N then ‘+’ is the binary operation in N.

For any two natural numbers (>0) 4,8 є N We have 4+8=12 є N then “+” is a binary operation in N whereas “-” is not a binary operation in N since for any two numbers 6,12 є N we have 6-12 = -6 does not belongs to N. i.e “-”’ is not a binary operation in N as (a,b) є N does not imply a-b є N.

ASSOCIATIVE LAW ‘*’ is a binary operation in S , for any a, b, c є S if (a*b)*c = a*(b*c) then the operation “*” is called ASSOCIATIVE in ‘S’. Ex: a, b, c є Z a+(b + c) = (a + b)+c let us take a=1, b=2, c=3 1+(2+3)= (1+2)+3 1+5= 3+3 6=6

Existence of Identity : In a non-empty set G for any aєG there exist e єG Э a*e = e*a = a then ‘e’ is called an Identity element in G w.r.to the operation “*”. → With respect to addition ‘0’ is the identity element since 0+a = a+0 = a. Ex : for any number -6 є Z there exists 0 є Z such that -6 + 0 = 0+ (-6) = -6 Ex : 3+0 = 0+3 = 3 → Under multiplication ‘1’ is the identity element

Inverse property : For any a є G there exists b є G Э a. b = b Inverse property : For any a є G there exists b є G Э a*b = b*a = e then ‘b’ is called Inverse element of ‘a’ in G. → Under addition inverse of a is –a → Under multiplication inverse of a is 1/a Ex : for 3є Z there exists -3 є Z Э 3+ (-3) = -3 + 3 = 0

Algebraic structure A non-empty set G equipped with one or more binary operations is called Algebraic structure. Ex: (R, .,+), (Z, .)

Semi Group An Algebraic structure which is Associative is called Semi group Ex:(N,+) is Semi group i . e, For a, b, c ЄN 1.aєN, bєN =>a + b є N 2.(a + b)+c = a+(b + c)

Monoid A Semi group with the identity element is known as “Monoid”. i Monoid A Semi group with the identity element is known as “Monoid” . i.e., (S,0) is a Monoid if a non-empty S under the operation “0” follows 1.Binary operation (or) Closure law 2. Associativity 3. Identity Ex: 1. (N,+) is not a monoid since 0 does not є N 2. (R,+) , (N,.) ,(Z,+) are the monoids. .

Group Let G be a non empty set and ‘ Group Let G be a non empty set and ‘*’ is a binary operation defined on G then (G,*) is said to be a Group if it satisfies the following properties 1. Associative property 2. Identity property and 3. Inverse property Ex : (Z, +), (R, +), (Q, +) etc are groups under “+”.

ABELIAN GROUP In a group (G,0) If a,b є G => a0b = b0a then (G,0) is called an “Abelian or Commitative group” . Ex: (Z,+), (R,+), (C,+), (Q,+) are the Commitative groups .

FINITE AND INFINITE GROUP In group (G,0) If the number of the elements are finite then it is called “Finite group”. Otherwise it is “Infinite group”. Ex: (R,+), (Z,+), (Q,+)

ORDER OF A GROUP The number of elements in a group (G,0) is called it’s “order”. And it is denoted by 0(G) . 0(G) < ∞. Ex: 0(G) = 2n for all n є Z is the group of even order. 0(G) = 2n-1 for all n є N is the group of the odd order.

Any subset H of a group G is called a Complex. Ex: Set of all even integers is complex of group (Z, +) Ex 2: In the Group, fourth roots of unity {1, -1, i, -i }, the complexes are {1, -1}, {i, -i}, …… 16

If a complex H of a group G is itself a group under the operation of G, we say that H is a Subgroup of G denoted by H≤G Ex: (Z, +) is the subgroup of (Q, +) (R, +) is the subgroup of (C, +)

If a complex H of a group G is itself a group under the operation of G, we say that H is a Subgroup of G denoted by H≤G Ex: (Z, +) is the subgroup of (Q, +) (R, +) is the subgroup of (C, +) If If

The fourth roots of unity {1, -1, i, -i} is the group G under multiplication. The complex {1, -1} is a subgroup but {i, -i} is not a subgroup WHY?

Criteria of a subset to be a subgroup: Theorem: H is a non-empty complex of a subgroup G, the necessary and sufficient condition for H to be a subgroup of G is a, b ϵ H =>ab^(-1) ϵ H  In additive notation, H is a subgroup of G iff a,b ϵ H => a-b ϵ H

Two-Step Subgroup Test Theorem: A non-empty complex H of a group G is a subgroup  1) a ϵ H, b ϵ H => ab ϵ H. 2) a ϵ H => a^(-1) ϵ H.

Finite Subgroup Test Theorem: Let H be a non-empty finite complex of a group G. If H is closed under the operation of G, then H is a subgroup of G. i.e. a, b ϵ H => ab ϵ H.

Some Other Criterions  A necessary and sufficient condition for a non-empty complex H of a subgroup G to be a subgroup of G is that HH^(-1) is contained in H.  A necessary and sufficient condition for a non-empty subset H of a subgroup G to be a subgroup of G is that HH^(-1) = H.

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