1. Prepared By Meri Dedania (AITS) Discrete Mathematics by Meri Dedania Assistant Professor MCA department Atmiya Institute of Technology & Science Yogidham Gurukul Rajkot

2. Prepared By Meri Dedania (AITS) Group Theory

3. Prepared By Meri Dedania (AITS) Definition of Group A Group is an algebraic system in which  on G satisfies four condition  Closure Property For all x, y  G x  y  G  Associative Property For all x, y, z  G x  (y  z) = (x  y)  z  Existence of Identity element There exists an element e  G such that for any a  G x  e = x = e  x  Existence of Inverse Element For every x  G,there exists an element denoted by a -1  G such that x -1  x = x  x -1 = e

4. Prepared By Meri Dedania (AITS) Definition of abelian Group A Group in which the operation  is commutative is called abelian Group i.e.  a,b  G, a  b = b  a Example 1. is Abelian Group 2. is abelian Group

5. Prepared By Meri Dedania (AITS) Theorem 1 : Let e be an identity element in group, Then e is unique Proof :  Let e and e` are two identity in G  e e` = e if e` is identity  e e` = e` if e is identity  since ee` is unique element in G  e = e` Properties of Group

6. Prepared By Meri Dedania (AITS) Theorem 2 : Inverse of each element of a group is unique Proof :  Let a be any element of G and e the identity of G  Suppose b and c are two different inverse of a in G.  a  b = e = b  a (if b is an inverse of a)  a  c = e = c  a (if c is an inverse of a)  Now, b = b  e = b  ( a  c) = (b  a)  c = e  c = c Thus a has unique inverse

7. Prepared By Meri Dedania (AITS) Theorem 3 : if a -1 is the inverse of an element a of group then (a -1)-1 =a Proof :  Let e be the identity of Group  a -1  a = e  (a -1 ) -1  (a -1  a) = (a -1 ) -1  e  ((a -1 ) -1  a -1 )  a = (a -1 ) -1  e  a = (a -1 ) -1  (a -1 ) -1 = a

8. Prepared By Meri Dedania (AITS) Theorem 4 : If be a group then for any two elements a and b of prove that ( a  b ) -1 = b -1  a -1 rule of reversal Proof :  Let a -1 and b -1 are inverse of a and b respectively and e be the identity  a  a -1 = e = a -1  a  b  b -1 = e = b -1  b  (a  b)  (b -1  a -1 ) = [(a  b)  b -1 ]  a -1  = [a  (b  b -1 )]  a -1  = [a  e]  a -1  = a  a -1  = e  Similarly, (b -1  a -1 )  (a  b) = e  This show that b -1 and a -1 is inverse of b and a  Hence, ( a  b ) -1 = b -1  a -1

9. Prepared By Meri Dedania (AITS) Cancellation Property : if a, b and cbe any three elements of a group then ab = ac  b = c left cancellation ba = ca  b = c right cancellation Proof :  Let a  G and also a -1  G  aa -1 = e = a -1 a  where e is identity of G  Now, ab = ac  a -1 (ab) = a -1 (ac)  (a -1 a) b = (a -1 a)c  e. b = e. c  b =c  similarly, ba = ca  b = c

10. Prepared By Meri Dedania (AITS) Definition of Permutation  A permutation is one to one mapping of non empty set P, say onto itself  Example : Let S = {1,2,3} Then function f : S  S f(1) = 2 f(2) = 3 f(3) = 1 Then permutation P1 = P2 = Permutation Group

11. Prepared By Meri Dedania (AITS) P3 = P4 = P5 = P6 = There are n! of pattern of expressing Permutation. So if Set has 3 elements then pattern of expressing permutation is 3! = 6

12. Prepared By Meri Dedania (AITS) Equality of Permutations :  Let f and g be two permutations defined on a non empty set P. Then f = g if and only if f(x) = g(x)  x  P  Example 1) Let S = {1,2,3,4} and let permutation f and g are equal or not.. f = g = 2) Let S = {1,2,3,4} and let permutation f and g are equal or not.. f = g =

13. Prepared By Meri Dedania (AITS) Permutation Identity  An Identity permutation on S, denoted by I, is defined as I(a) = a  a  S  For example : f = Note : In identity permutation the image of element is element itself

14. Prepared By Meri Dedania (AITS) Composition of Permutation ( Product of Permutation)  Let f and g be two arbitrary permutations of like degree, given by, f = g = on non empty set A. Then the composition (or Product) of f and g is defined as Continue…

15. Prepared By Meri Dedania (AITS) f  g =  =  Example Let P 1 = P 2 = P 3 = Check P 1  (P 2  P 3 ) = (P 1  P 2 )  P 3

16. Prepared By Meri Dedania (AITS) Inverse Permutation  Every permutation f on set P = {a 1,a 2,a 3,…,a n } Possesses a unique inverse permutation, denoted by f -1 thus if f = Then f -1 =

17. Prepared By Meri Dedania (AITS) Cyclic Permutation  Let t 1,t 2,…..,t r be r distinct elements of the set P = {t 1,t 2,…., t n }.Then the permutation p : P  P is defined by p(t 1 ) = t 2, p(t 2 ) = t 3,….,p(t r -1)= t r, p(t r )=t 1 is called a cyclic permutation of length r.  Example : The permutation P = is written as (1,2), (3,4,6), (5).. The cycle (1,2) has length 2, The cycle length 3,The cycle 1.

18. Prepared By Meri Dedania (AITS) Definition of Cyclic Group  If there exists an element a  G for some group such that every element of G can be written as some power of a, that is a n for some integer n. then a Group is said to be cyclic Group  Every Cyclic Group is abelian  Example for set A = { , , ,  } and binary operation 

19. Prepared By Meri Dedania (AITS) I. If is an abelian group, then for all a, b  G show that ( a  b ) n = a n  b n Solution ( a  b ) n = a n  b n ( a  b ) n+1 =a n+1  b n+1 ( a  b ) n+2 =a n+2  b n+2 Now, ( a n  b n ) ( a  b ) = ( a  b ) n+1 = ( a n+1  b n+1 )  (b n  a )=(a  b n ) By cancellation, similarly b n+1  a = a  b n+1 Again b n+1  a = b(b n  a) = b(ab n ) i.e., ab n+1 = b(ab n )

20. Prepared By Meri Dedania (AITS) III. Show that in a Group, if for any a, b  G, ( a  b ) 2 = a 2  b 2, then must be abelian Solution : Let be a Group and let a, b  G ( a  b ) 2 = a 2  b 2  ( a  b )  ( a  b ) = ( a  a)  ( b  b )  a  ( b  a)  b = a  ( a  b)  b  By left and right cancellation property  b  a = a  b  Thus we have a  b = b  a.  a,b  G  Hence is an abelian Group

21. Prepared By Meri Dedania (AITS) IV. Show that if every element in a group is its own inverse, then the group must be abelian Solution : Let a, b  G  a  b  G (by closure property) Now, a -1 = aand b -1 = b  ( a  b) -1 = a  b Now, ( a  b) -1 = a  b  b -1  a -1 = a  b  b  a = a  b  Thus we have a  b = b  a,  a,b  G  Hence is an abelian Group

22. Prepared By Meri Dedania (AITS) V. Write down Composition table for and where Z 7 * = Z 7 - {0} + 7 0123456 00123456 11234560 22345601 33456012 44560123 55601234 66012345 77 123456 1123456 2246135 3362514 4415263 5531642 6654321

23. Prepared By Meri Dedania (AITS) VII. Show that and are the only finite groups of nonzero real numbers under the operation of multiplication Solution:

24. Prepared By Meri Dedania (AITS) Definition of Sub Group :  Let be a Group and S  G, such that it satisfies the following condition: 1) e  G, Where e is the identity of 2) For any a  S, a-1  S 3) For a, b  S, a  b  S Then is called Sub Group of  For any group, and are Trivial Sub Groups of.  Let is a Group then & are Trivial Sub group of  All other subgroups of are called Proper Subgroup  Let is Proper subgroup of Sub Group and Homomorphism

25. Prepared By Meri Dedania (AITS) Theorem : A subset S   of G is a subgroup of iff for any pair of elements a, b  S, a  b-1  S Proof :  Assume that S is a subgroup  if a, b  S then b -1  S and a  b -1  S  To prove the converse, let us assume that a, b  S and a  b -1  S for any pair a, b.  taking b = a, a  a -1 = e  S  From e, a, b  S e  a -1 = a -1  S  Similarly, b -1  S.  Finally, because a and b -1 are in S, we have a  b  S.  Hence, is a sub group of

26. Prepared By Meri Dedania (AITS) Definition of Group Homomorphism  Let and be two Group. A mapping g : G  H is called a group homomorphism from to if for any a, b  G  g (a  b) = g(a)  g(b)  g(e G ) = e H  g(a -1 ) = [g(a)]-1

27. Prepared By Meri Dedania (AITS) Definition of Group Isomorphism  Let f : .if f is one to one and onto. Then Group is called isomorphism  A homomorphism f :  is called an endomorphism  A Isomorphism f :  is called an automorphism

28. Prepared By Meri Dedania (AITS) Definition Kernal of Homomorphism  Let and be two Groups and let f is homomorphism of G into H. The set of elements of G which are mapped into e H, the identity of H is called the kernal of the homomorphism and is denoted by K f or Ker(f)

29. Prepared By Meri Dedania (AITS) Theorem : The Kernal of homomorphism f :  is sub group of Proof :  Here f :  is homomorphism  Ker (f) = {x  G | f(x) = e H identity element of H}  k (f)   because e G  K(f) (f(e G )=e H )  let a, b  K f  f (a) = e H & f(b) = e G  Now, f(ab -1 ) = f(a). f(b -1 )  = f(a). [f(b)] -1  = e H. e H -1  = e H. e H  = e H  ab -1  K f  K f is a sub group of

30. Prepared By Meri Dedania (AITS) Show that every interval of lattice is a sub lattice of a lattice Proof:  Let be a lattice and a, b  L  a, b   a  a  b  a  [ a, b ]  also [a, b] = {x  L | a  x  b}  L  Let x, y  [ a, b ]  a  x  b, a  y  b  a  a  x  y  b  b  a  a  x  y  b  b  a  x  y  b  a  x  y  b  x  y  [ a, b ] and x  y  [ a, b ]  [ a, b ] is sub lattice of the lattice

31. Prepared By Meri Dedania (AITS) Draw Hasse Diagram of the poset {2,3,5,6,9,15,24,45},D. Find (i) Maximal and Minimal elements (ii) Greatest and Least members, if exist. (iii) Upper bound of {9,15} and l.u.b. of{9,15}, if exist. (iv) Lower bound of {15,24} and g.l.b. of{15,24}, if exist.

32. Prepared By Meri Dedania (AITS) Definition Of Right Cosets  Let G be a Group and H is any sub Group of G. Let a be any element of G. Then set Ha = {ha : h  H} is called a right coset of H in G generated by a. Definition of Left Cosets  Let G be a Group and H is any sub Group of G. Let a be any element of G. Then set aH = {aH : h  H} is called a right coset of H in G generated by a.

33. Prepared By Meri Dedania (AITS) Lagrange’s Theorem :  The order of each sub group of a finite group G is a divisor of the order of G  Index in G :  The number of left cosets of H in G is called index of H in G.  Definition of Normal subgroup:  A sub group is sub group of is called a normal sub group if for any a  G, aH = Ha

34. Prepared By Meri Dedania (AITS) Fins the Sub group of show that is subgroup of Show that every sub group of abelian group is normal  Let x  G and h  H  Xhx -1 = xx -1 h = eh = h   x  G and  h  H  xhx -1  H  i.e. xH = Hx  H is normal subgroup of G

35. Prepared By Meri Dedania (AITS) Definition of Irreflexive  A relation R on a set A is irreflexive if aRa for  a  A, if (a,a)  R  For example  A = {1,2,3,4}  R = {,,,, }  R = {, }