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Group algebra This is the same idea as matrix algebra in FP1.

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

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Group algebra This is the same idea as matrix algebra in FP1.

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  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. Ex2 Prove (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 = a hence using a • b = a + b – ab a• e = a + e – aereplace b by e a • e = a + e – aas 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 x2 = e x  X Now look at x •y: x • y = e •( x • y )• e as this changes nothing by identity = y • y • x • y • x • x as x • x = y • y = e for this group = y • (y • x) • (y • x) • x by associativity = y • e • x as (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 = y2 x prove that x y  y x Proof Assume xy = y x so y x = y2 x = y y x Post multiply by (yx)–1 (y x)(yx)–1 = y (y x) (yx)–1 e = y e = y but y  e so x y  y x

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