A GROUP has the following properties:

1. A GROUP has the following properties: • Closure • Associativity • Identity • every element has an Inverse

2. G = { i, k, m, p, r, s } is a group with operation * as defined below: G has CLOSURE: for all x and y in G, x*y is in G. The IDENTITY is i : for all x in G, ix = xi = x Every element in G has an INVERSE: k*m = i p*p = i r*r = i s*s = i

3. G has ASSOCIATIVITY: for every x, y, and z in G, (x*y)*z = x*(y*z) for example: ( k*p )* r ( s )* r m = k* ( p* r ) k* ( k ) m

4. G = { i, k, m, p, r, s } is a group with operation * as defined below: G does NOT have COMMUTATIVITY: p*r = r*p

5. H = { i, k, m } is a SUBGROUP

6. H = { i, k, m } is a SUBGROUP definition: If G is a group, H is a subgroup of G, and g is a member of G then we define the left coset gH: gH = { gh / h is a member of of H }

7. H = { i, k, m } is a SUBGROUP Example: to form the coset r H definition: If G is a group, H is a subgroup of G, and g is a member of G then we define the left coset gH: gH = { gh / h is a member of of H }

8. H = { i, k, m } is a SUBGROUP r r r r Example: to form the coset r H H = { i , k , m } definition: If G is a group, H is a subgroup of G, and g is a member of G then we define the left coset gH: gH = { gh / h is a member of of H }

9. H = { i, k, m } is a SUBGROUP r r r r Example: to form the coset r H H = { i , k , m } r s p = { r , s , p } definition: If G is a group, H is a subgroup of G, and g is a member of G then we define the left coset gH: gH = { gh / h is a member of of H }

10. H = { i, k, m } = a subgroup The COSETS of H are: iH = { i*i, i*k, i*m }={i,k,m} kH = { k*i, k*k, k*m }={k,m,i} mH = {m*i,m*k, m*m}={m,i,k} pH = { p*i, p*k, p*m }={p,r,s} rH = { r*i, r*k, r*m }={r,s,p} sH = { s*i, s*k, s*m }={s,p,r}

11. The cosets of a subgroup form a group: A B A A B BB A

12. M = { A,B,C,D,E,F,G,H } is a noncommutative group. N = { B, C, E, G } is a subgroup of M

14. The cosets of N = { B, C, E, G } are: AN = { D,F,A,H} BN = { C,G,B,E} CN = { G,E,C,B} DN = { F,H,D,A} EN = { B,C,E,G} FN = { H,A,F,D} GN = { E,B,G,C} HN = { A,D,H,F}

15. Rearrange the elements of the table so that members or each coset are adjacent and see the pattern!

16. Q is a commutative group R = { c, f, I } is a subgroup of Q

17. The cosets of R: { d,g,a } { e,h,b,} { c,f,I }

18. The cosets of a subgroup partition the group: ie: every member of the group belongs to exactly one coset. a b c d e f g h i LAGRANGE’S THEOREM: the order of a subgroup is a factor of the order of the group. (The “order” of a group is the number of elements in the group.)

19. If we rearrange the members of Q, we can see that the cosets form a group

22. familiar examples of groups:

23. Z T example 1: the INTEGERS with the operation + closure: the sum of any two integers is an integer. associativity: ( a + b ) + c = a + ( b + c ) identity: 0 is the identity every integer x has an inverse -x {………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………} The multiples of three form a subgroup of the integers: {………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………} With coset: (add 1 to every member of T) {………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………}

24. Z T example 1: the INTEGERS with the operation + closure: the sum of any two integers is an integer. associativity: ( a + b ) + c = a + ( b + c ) identity: 0 is the identity every integer x has an inverse -x {………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………} The multiples of three form a subgroup of the integers: {………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………} With coset: (add 1 to every member of T) {………-5,-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………} and coset (add 2 to every member of T)

25. R2 = example 3: The set of points on a line through the origin is a SUBGROUP of R2. eg: y = 2x If the vector is added to every point on y = 2x You get a coset of L example 2: The set of all points on the plane with operation + defined: The identity is the origin. L=

26. No element is repeated in the same row of the table. No element is repeated in the same column of the table. Theorem: Every group has the cancellation property.

27. No element is repeated in the same row of the table. No element is repeated in the same column of the table. Theorem: Every group has the cancellation property. Because r is repeated in the row, if a x = a y you cannot assume that x = y . In other words, you could not “cancel” the “a’s”

28. No element is repeated in the same row of the table. No element is repeated in the same column of the table. If then Theorem: Every group has the cancellation property. In a group, every element has an inverse and you have associativity.

29. EXERCIZE

30. COMPLETE THE TABLE TO MAKE A GROUP:

31. What is the IDENTITY? If r were the identity, then rw would be w If s were the identity, then sv would be v If w were the identity, then wr would be r

32. The IDENTITYis t tr = r r

33. The IDENTITYis t tr = r ts = s s r

34. tt = t tu = u t u v w tv = v tw = w The IDENTITYis t tr = r ts = s s r

35. rt = r st = s r tt = t s ut = u t vt = v u wt = w v w The IDENTITYis t and s r u v w

36. sv = t s and v are INVERSES vs = t t

37. u is its own inverse

38. INVERSES: sv = t tt = t uu = t What about w and r ? t w and r are not inverses. w w = tand rr = t t

39. CANCELLATION PROPERTY: no element is repeated in any row or column u and w are missing in yellow column There is a u in blue row u uv must be w rv must be u w

40. u and v are missing in yellow column There is a u in blue row uw must be v vw must be u v u

41. r and s are missing r and w are missing s and u are missing u is missing u r s w r u s

42. Why is the cancellation property useless in completing the remaining four spaces? v and w are missing from each row and column with blanks. We can complete the table using the associative property.

43. ASSOCIATIVITY ( r s ) w = r ( s w ) ( r s ) w = r ( s w ) ( r s ) w = r ( r ) ( r s ) w = t ( r s ) w = t w w

44. w v w v