GROUP Closure Associativity Identity A group is a mathematical system that has the following properties: Closure Associativity Identity every element has an Inverse

example 1

G = { 1, 2, 3, 4 } operation defined: x y = the remainder when xy is divided by 5  1 2 3 4

G = { 1, 2, 3, 4 } operation defined: x y = the remainder when xy is divided by 5  1 2 3 4 3 4 = the remainder when 12 is divided by 5

G = { 1, 2, 3, 4 } operation defined: x y = the remainder when xy is divided by 5  1 2 3 4 The system has CLOSURE

G = { 1, 2, 3, 4 } operation defined: x y = the remainder when xy is divided by 5  1 2 3 4 The system has an IDENTITY = 1

G = { 1, 2, 3, 4 } operation defined: x y = the remainder when xy is divided by 5  1 2 3 4 The system has an IDENTITY = 1 The INVERSE of 2 is 3 The INVERSE of 4 is 4

G = { 1, 2, 3, 4 } operation defined: x y = the remainder when xy is divided by 5  1 2 3 4 The system has ASSOCIATIVITY 4  ( 2  3 ) (4  2 )  3

G = { 1, 2, 3, 4 } operation defined: x y = the remainder when xy is divided by 5  1 2 3 4 The system has ASSOCIATIVITY 4  ( 2  3 ) (4  2 )  3 4  ( 1 ) (3 )  3

G = { 1, 2, 3, 4 } operation defined: x y = the remainder when xy is divided by 5  1 2 3 4 The system has ASSOCIATIVITY = 4  ( 2  3 ) (4  2 )  3 4  ( 1 ) (3 )  3 4 4 etc.

example 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

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

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

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

example 3

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

example 4

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

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

EXERCIZE

COMPLETE THE TABLE TO MAKE A GROUP:

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

The IDENTITY is t tr = r r

The IDENTITY is t tr = r ts = s r s

tr = r ts = s tt = t tu = u tv = v tw = w The IDENTITY is t t u r s w

rt = r st = s tt = t ut = u vt = v wt = w r s u v w The IDENTITY is t and rt = r r st = s s tt = t t ut = u u r s u v w vt = v v wt = w w

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

u is its own inverse

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

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

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

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

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.

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

w v w v