1. Group algebra This is the same idea as matrix algebra in FP1. Ex1 a b = c make b the subject a -1 on left: a -1 (a b) = a -1 c use associative rule (a -1 a) b = a -1 c use inverse rule e b = a -1 c use identity rule b = a -1 c

2. Ex2Prove (a b c) -1 = c -1 b -1 a -1 Use inverse rule (a b c) -1 (a b c) = e c -1 on right: (a b c) -1 a b c c -1 = e c -1 inverse and identity rules: (a b c) -1 a b e = e c -1 (a b c) -1 a b = c -1 b -1 on right: (a b c) -1 a b b -1 = c -1 b -1 inverse and identity rules: (a b c) -1 a e = c -1 b -1 (a b c) -1 a = c -1 b -1 a -1 on right: (a b c) -1 a a -1 = c -1 b -1 a -1 inverse and identity rules: (a b c) -1 e = c -1 b -1 a -1 (a b c) -1 = c -1 b -1 a -1

3. Ex3 Prove that if c commutes with every element of a group then c -1 does too. Let x be any element in the group. We are told c x = x c and the objective is to prove that c -1 x = x c -1 c x = x c c -1 on right left and c -1 on right : c -1 c x c -1 = c -1 x c c -1 e x c -1 = c -1 x e x c -1 = c -1 x as required

4. Ex4 Find an inverse of a in R–{1} with the binary operation a b = a + b – ab R–{1}means the set of real numbers excluding 1 1) Find the identity element a e = ahence usinga b = a + b – ab a e = a + e – aereplace b by e a e = a + e – a as ae = a a e = e ae – e = 0 e(a – 1) = 0 as a is an element in R–{1} a  1 So e = 0

5. Self-inversive groups A system of algebra exists for every group, which can be used to make surprising conclusions: e.g.If a group X is self - inversive then x 2 = e x  X Now look at x y: x y = e ( x y ) eas this changes nothing by identity = y y x y x xas x x = y y = e for this group = y (y x) (y x) xby associativity = y e xas (y x)  X by closure = y x Conclusion: Self-inversive groups are commutative

6. Using contradiction Ex1 If x, y and the identity element e are elements of a group H and x y = y 2 x prove that x y  y x ProofAssume x y = y x so y x = y 2 x = y y x Post multiply by (y x) –1 (y x) (y x) –1 = y (y x) (y x) –1 e = y e = y but y  eso x y  y x