Understanding Algebraic Groups: Definition, Properties, and Examples

1. GROUPS

2. 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.

3. 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 a set to give another elements of the same set is called a binary operation

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

5. 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.

6. Definition of Groups A group (G, ・) is a set G together with a binary operation ・ satisfying the following axioms. • 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 a-1・ a = a ・ a−1 = e.

7. 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

8. 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.

9. 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.

10. EXAMPLE (2)Show that Z(the set of all integers) is an abelian group w.r.t. addition Solution. • 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

11. 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.

12. (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.

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

14. 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.

15. 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

16. 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.

17. 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).

18. 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.

19. The Identity Element Of G Is Unique Proof:: If possible let e1 and e2 be two identities in the group (G, .) Since e1 is identity and e2 ∈ G therefore e1 . e2 = e2 = e2. e1 …..(1) Also since e2 is the identity and e1 ∈ G therefore e1 . e2 = e1 = e2 . e1 …..(2) therefore from (1) and (2), e1 = e2 Hence the identity is unique

20. Every element has unique inverse PROOF: Let a be any element of the group ( G , .) If possible , let b1 and b2 be two inverses of a under the binary operation ‘ . ‘ and let e be the identity element in G. Then a . b1 = e = b1 . a And a . b2 = e = b2 . a Now b1 = b1 . e =b1 . ( a . b2)

21. b1=(b1 . a) . b2 [by associativity] =e . b2 = b2 Therefore b1 = b2. hence the inverse is unique.

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

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

24. Proof of Proof:

25. SUBGROUPS

26. Definition of Subgroup • 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 ∈ H. (closure) (ii) a−1 ∈ H for all a ∈ H. (existence of inverses) • Conditions (i) and (ii) are equivalent to the single condition: (iii) a ・ b−1 ∈ H for all a, b ∈ H.

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

28. 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 As H is a sub-set of G therefore ea = ae= a for all a∈ H [ since a∈ H => a∈ G] => e is an identity element of H. Hence identity of the sub-group is same as that of the group.

29. Proof:- (ii) Let e be the identity of G as well as of H. Let a be any element of H • 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. • ab = ba = e ….. (1) ac = ca = e …. (2) From (1) & (2), ab = ac Therefore b=c Hence, the result.

30. 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 = {… , -3k , -2k , -k , 0 , k , 2k , 3k ,….} Therefore H ≠ ф Here H is a subset of G. We have to show that H is a subgroup of G.

31. 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.

32. 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.

33. 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 hasinfinite order.

34. CYCLIC GROUP AND COSETS

35. CYCLIC GROUPS • Definition. 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.

36. 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.

37. 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 = 1+ 6 1 = 2 1 = 1 + 6 1 = 3 1 = 1 +6 1 = 4 1 2 3 2 4 3

38. 1 = 1 + 6 1 = 5 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 5 4 5 6

39. 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 aH = { ah : h  H} is called left coset of H in G generated by a. Obviously , Ha and aH are both subsets of G . H is itself a right and left coset as eH = H =He , where e is an identity of G.

40. 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

41. 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.

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

43. ASSIGNMENT 4. If a, b are elements of a group G , then (i) (a−1)−1 = a. (ii) (ab)−1 = b−1a−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

44. 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.

45. 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?

46. ASSIGNMENT 12.Let G be a group with binary operation denoted as multiplication. The set {h ∈ G : for all x ∈ G} is called 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.

47. 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.

48. 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−1a−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.

49. 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.