GROUPS 1 1 Introduction Detailed Study of groups

GROUPS 1

1. Introduction Detailed Study of groups is a fundamental concept in the study of abstract algebra. To define the notion of groups, we require the concept of binary operation or composition which is a type of function that associates two elements of the set to a unique element of that set. 2

Definition Of Binary Operation A binary operation on a set is a rule for combining two elements of the set. More precisely, if S iz a nonempty set, a binary operation on S iz a mapping f : S S S. Thus f associates with each ordered pair (x, y) of element of S an element f(x, y) of S. IN OTHER WORDS, An operation which combine two elements of a set to give another elements of the same set is called a binary operation 3

Example Of Binary Operation 1. Ordinary addition ‘+’ is a binary operation on Z Consider +: (Z*Z) Z +: (3, 7)=3+7=10 ∈Z 4

Definition Of Algebraic Structure A non-empty set G equipped with one or more binary operation defined on it is called an algebraic structure. Suppose ‘*’ is a binary operation on G , then (G , *) is an algebraic structure. 5

Definition of Groups A group (G, ・) is a set G together with a binary operation ・ satisfying the following axioms. (i) Closure property: a. b ∈ G for all a, b ∈ G. (ii) The operation ・ is associative; that is, (a ・ b) ・ c = a ・ (b ・ c) for all a, b, c ∈ G. (iii) There is an identity element e ∈ G such that e ・ a = a ・ e = a for all a ∈ G. (iv) Each element a ∈ G has an inverse element a− 1 ∈ G such that -1 a・ a = a ・ a− 1 = e. 6

Abelian Group A group (G , . ) is said to be abelian or commutative if in addition to the above four postulates, the following postulate is also satisfied: COMMUTATIVE PROPERTY: If the operation is commutative, that is, if a ・ b = b ・ a for all a, b ∈ G, the group is called commutative or abelian group 7

Example Of Group Example: Let G be the set of complex numbers {1, − 1, i, −i} and let ・ be the standard multiplication of complex numbers. Then (G, ・) is an abelian group. The product of any two of these elements is an element of G; thus G is closed under the operation. Multiplication is associative and commutative in G because multiplication of complex numbers is always associative and commutative. 8

Example Of Group (CONTD. ) The identity element is 1, and The inverse of each element a is the element 1/a. Hence 1− 1 = 1 , (− 1)− 1 = − 1 , i− 1 = −I , and (−i)− 1 = i. 9

EXAMPLE (2) Show that Z(the set of all integers) is an abelian group w. r. t. addition Solution. (i) CLOSURE PROPERTY: since the sum of two integers is also an integer i. e. , a+b ∈ Z for all a, b ∈ Z Therefore the set Z is closed w. r. t. addition. Hence closure property is satisfied. (ii) ASSOCIATIVE LAW: since addition of integers obey associative law, therefore a+(b+c)=(a+b)+c for all a, b, c ∈ Z 10

Thus addition is an associative composition. (iii) EXISTENCE OF IDENTITY: the number 0 ∈ Z and a+0 = 0+a =a for all a ∈ Z The integer 0 is the identity for (Z , +) (iv) EXISTENCE OF INVERSE: for each a ∈ Z, There exists a unique element -a ∈ Z such that a+(-a)=0=(-a)+a Thus each integer possesses an additive inverse. 11

(v) COMMUTATIVE LAW: the commutative law holds good for addition of integers i. e. a+b=b+a for all a, b ∈ Z Thus (Z , +) is an abelian group. Also Z contains an infinite number of elements. therefore (Z , +) is an abelian group of infinite order. 12

Definition Of Semi-Group An algebraic structure (G , *) is called a semigroup, if only the first two postulates, i. e. , closure and associative law are satisfied. 13

Example Of Semi-Group The algebraic structure (N , +), (W , +), (Z, +), (R, +) and (C, +) are semi-groups, where N , W , Z , R and C have usual meanings. 14

Definition Of Finite And Infinite Groups Definition: If the number of elements in the group G are finite, then the group is called a finite group otherwise it is an infinite group 15

Example Of Finite And Infinite Group Example of Finite Group: {1, − 1, i, −i} is an example of finite group. Example of Infinite Group: (Z , +) is an example of infinite group. 16

Order Of A Group Definition: The number of elements in a finite group is called the order of the group. An infinite group is said to be of infinite order. The order of a group G is denoted by the symbol o(G). 17

General properties of groups If (G , . ) is a group, then (i) the identity element of G is unique. (ii) every element has a unique inverse. 18

The Identity Element Of G Is Unique Proof: : If possible let e 1 and e 2 be two identities in the group (G, . ) Since e 1 is identity and e 2 ∈ G therefore e 1. e 2 = e 2. e 1 …. . (1) Also since e 2 is the identity and e 1 ∈ G therefore e 1. e 2 = e 1 = e 2. e 1 …. . (2) therefore from (1) and (2), e 1 = e 2 Hence the identity is unique 19

Every element has unique inverse PROOF: Let a be any element of the group ( G , . ) If possible , let b 1 and b 2 be two inverses of a under the binary operation ‘. ‘ and let e be the identity element in G. Then a. b 1 = e = b 1. a And a. b 2 = e = b 2. a Now b 1 = b 1. e =b 1. ( a. b 2) 20

b 1=(b 1. a). b 2 [by associativity] =e. b 2 = b 2 Therefore b 1 = b 2. hence the inverse is unique. 21

Proposition. If a, b are elements of a group G , then (i) (a− 1)− 1 = a. (ii) (ab)− 1 = b− 1 a− 1. i. e. , the inverse of the product of two elements of a group is the product of their inverses in the reverse order. 22

Proof of (a− 1)− 1 = a Proof: for each a ∈ G, we have a. a− 1 = e = a− 1. a Þ Inverse of a− 1 is a. Therefore (a− 1)− 1 = a 23

Proof of Proof: 24

SUBGROUPS 25

Definition of Subgroups: : It often happens that some subset of a group will also form a group under the same operation. Such a group is called a subgroup. If (G, ・) is a group and H is a nonempty subset of G, then (H, ・) is called a subgroup of (G, ・) if the following conditions hold: (i) a ・ b ∈ H for all a, b ∈closure) H. ( (ii) a− 1 ∈ H for all a ∈ existence H. ( of inverses) Conditions (i) and (ii) are equivalent to the single condition: (iii) a ・ b− 1 ∈ H for all a, b ∈ H. 26

THEOREM: Prove that: (i) The identity of the sub-group is same as that of the group. (ii) The inverse of any element of a sub-group is the same as the inverse of that element in the group. 27

Proof : let H be a sub-group of the group G. if e is the identity element of G, then ea = ae = a for all a ∈ G (i) As H is a sub-set of G therefore ea = ae= a for all a ∈ H ∈ G] [ since ∈ H => a a => e is an identity element of H. Hence identity of the sub-group is same as that of the group. 28

Proof: (ii) Let e be the identity of G as well as of H. Let a be any element of H Þ a is an element of group G suppose b is the inverse of a in H and c is the inverse of a in G. Þ ab = ba = e …. . (1) ac = ca = e …. (2) From (1) & (2), ab = ac Therefore b=c Hence, the result. 29

Example of subgroup: LET G be the additive group of integers. Prove that the set of all multiples of integers by a fixed integer k ig a subgroup of G. Solution. Let G = {…. , -4 , -3 , -2 , -1 , 0 , 1 , 2 , 3 , 4 , …. . } i. e. G is additive group of integers. Let H = {… , -3 k , -2 k , -k , 0 , k , 2 k , 3 k , …. } Therefore H ≠ ф Here H is a subset of G. We have to show that H is a subgroup of G. 30

Let ak , bk be any two elements of H such that a , b are integers. Inverse of bk in G is –bk. Now ak - bk = (a - b)k , which is an element of H as (a – b) is some integer. Thus for ak , bk ∈ H , we have ak – bk ∈ H Hence H is a subgroup of G. 31

Example: Let H be the multiplicative group of all positive real numbers and R the additive group of all real numbers. Is H a subgroup of R ? Solution : H is a subset of R but H is not a subgroup of R , The reason being that the composition in H is different from the composition in R. 32

Order of an element Definition. Let G be a group and let a G ane e be the identity element in G. If ak = e for some k 1, then the smallest such exponent k 1 is called the order of a; if no such power exists, then one says that a has infinite order. 33

CYCLIC GROUP AND COSETS 34

CYCLIC GROUPS If G is a group and a G, write <a > = {an : n Z} = {all powers of a }. It is easy to see that <a > is a subgroup of G. < a > is called the cyclic subgroup of G generated by a. A group G is called cyclic if there is some a G with G = < a >; in this case a is called a generator of G. Definition. 35

THEOREM: If a finite group of order ‘s’ contains an element of order ‘s’ , then the group must be cyclic. Proof: Let G be a finite group and o(G)= s Let a G such that o(a)= s If H = {an : n Z} Then o(H) = s = o(a) Therefore H is a cyclic subgroup of G. Also o(H) = o(G) implies G itself is a cyclic group and a is a generator of G. 36

EXAMPLE of cyclic group. Example: If G = {0 , 1 , 2 , 3 , 4 , 5 } and binary operation + 6 is Then prove that G is a cyclic group. Solution. we see that 1 1=1 12 = 1+ 6 1 = 2 3 2 1=1 +61=3 4 3 1 = 1 +6 1 = 4 37

5 4 1=1 +6 1=5 5 6 1 =1 +6 1=0 Therefore G = { 0 , 1 , 2 , 3 , 4 , 5 } is a cyclic group. 1 is generator of given group. Therefore G=<1> Hence proved 38

COSETS: Definition: let G be a group and H be any subgroup of G. For any a G , the set Ha = { ha : h H} is called right coset of H in G generated by a. Similarly, the set a. H = { ah : h H} is called left coset of H in G generated by a. Obviously , Ha and a. H are both subsets of G. H is itself a right and left coset as e. H =He , where e is an identity of G. 39

Example of cosets: Find the right cosets of the subgroup { 1, -1 } of the group { 1 , -1 , i, -i} w. r. t. usual multiplication. Solution. let G = { 1 , -1 , i, -i} be a group w. r. t. usual multiplication And H = { 1, -1 } be a subgroup of G. The right cosets of H in G are H(1) = {1(1) , -1(1)} = {1 , -1}= H 40

H(-1)={1(-1) , -1(-1)}={-1 , 1}=H H(i)={1(i) , -1 (i)} ={i, -i} H(-i) = {1(-i) , -1(-i)}= {-i , i} thus we have only two distinct right cosets of H in G. 41

ASSIGNMENT Define group. Show that Z (the set of all integers) is an abelian group w. r. t. addition. 2. Show that the set of integers Z is an abelian grop w. r. t. binary operation ‘ * ‘ defined as a * b = a + b + 1 for a , b ∈Z. 3. If (G , . ) is a group, then (i) the identity element of G is unique. 1. (ii) every element has a unique inverse. 42

ASSIGNMENT 4. If a, b are elements of a group G , then (i) (a− 1)− 1 = a. (ii) (ab)− 1 = b− 1 a− 1. i. e. , the inverse of the product of two elements of a group is the product of their inverses in the reverse order. 5. If every element of a group is its own inverse, then show that the group is abelian. 6. If a group has four elements , show that it must be abelian 43

ASSIGNMENT 7. The order of every element of a finite group is finite and is less than or equal to the order of the group. 8. Let G = { 0 , 1 , 2 , 3 , 4 , 5 }. Find the order of elements of the group G under the binary operation addition modulo 6. 9. Define subgroup. Let G be the additive group of integers. Prove that the set of all multiples of integers by a fixed integer k ig a subgroup of G. 44

ASSIGNMENT 10. Prove that: (i) the identity of the subgroup is same as that of the group. (ii) the inverse of any element of a subgroup is the same as the inverse of that element in the group. 11. Let H be the multiplicative group of all positive real numbers and R the additive group of all real numbers. Is H a subgroup of R? 45

ASSIGNMENT 12. Let G be a group with binary operation denoted as multiplication. The set {h ∈ G : for all x }∈is. Gcalled the centre of the group G. Show that the centre of G is a subgroup of G. 13. Define cyclic group. If a finite group ‘ s ‘ contains an element of order ‘ s ‘ , then the group must be cyclic. 46

ASSIGNMENT 14. If G = { 0 , 1 , 2 , 3 , 4 , 5 } and binary operation is addition modulo 6 , then prove that G is a cyclic group. 15. Define cosets. Find the right cosets of the subgroup { 1 , -1 } of the group { 1, -1 , i , -i} w. r. t. usual multiplication. 47

TEST DO ANY THREE QUESTIONS. 1. If a, b are elements of a group G , then (i) (a− 1)− 1 = a. (ii) (ab)− 1 = b− 1 a− 1. i. e. , the inverse of the product of two elements of a group is the product of their inverses in the reverse order. 2. Define cyclic group. If a finite group ‘ s ‘ contains an element of order ‘ s ‘ , then the group must be cyclic. 48

TEST 3. If G = { 0 , 1 , 2 , 3 , 4 , 5 } and binary operation is addition modulo 6 , then prove that G is a cyclic group. 4. Define cosets. Find the right cosets of the subgroup { 1 , -1 } of the group { 1, -1 , i , -i} w. r. t. usual multiplication. 5. Define subgroup. Let G be the additive group of integers. Prove that the set of all multiples of integers by a fixed integer k ig a subgroup of G. 49