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# 11 Math C - Introduction to Groups - tyoun158

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Tweet## Triston Young

on 3 March 2011#### Transcript of 11 Math C - Introduction to Groups - tyoun158

Introduction To Groups Modulo Arithmetic Modulo Arithmetic Not to be confused with modlus Orignally based on the clock but we don't worry about this. Modulo Arithmetic is like junior school math where we divide and have a remainder 22(mod4) = 2 What we are saying here is that when we divide 22 by 4 we have a remainder or residual of 2 _________________________________ A Cayley table is a tabulated set of results for modulo arithmetic Addition under Modulo 4 (mod4) Definitions of Terms Closure: Associativity: Identity: Inverse: a b = S 2 + 3 = 5 (2,3,5 C R) o An operation is closed if the result is of the same set as the equation - | Indicates any operation +, -, -, x NOTE: (a b) c = a (b c) o o o o An operation is associative if the order of the operation does not affect the answer a u = u a = a eg. 2 x 1 = 1 x 2 = 2 o o An equation has an identity if the operator leaves every element unchanged 1 is an identity

under multiplication a a = a a = u eg. 2 x 2 = 2 x 2 = 1 o o -1 -1 -1 -1 An equation has an inverse if the operation gives the identity A Further Observation is that: Commutativity: a b = b a o o An operation is commutative if the order of the elements doesn't effect the answer Exercise 4A All Parts Exercise 4B Even Q's Does [{mod 5},+] form an Abelian Group (mod 5), addition 1) Construct a Cayley Table for (mod 5)

2) Perform 5 Tests for Abelian group

Closed

Associative

Identity

Inverse

Commutative Cayley Table [{mod5},+] Tests for Abelian 1. Closure

2. Associativity

3. Does an Identity exist

4. Does an Inverse exist

5. Commutativity 1. All results are of the same set, so it the set is closed

2. Addition with whole numbers is associative

3. There exists a value of 0 in the Cayley Table (Identity)

4. There exists a value of 0 in every row (Inverse)

5. If there is a line of symmetry down the leading diagonal of a Cayley Table then the set is commutative

All 5 are true so [{mod5},+] is an Abelian Group Exercise 4C Q2,4,5-9

Full transcriptunder multiplication a a = a a = u eg. 2 x 2 = 2 x 2 = 1 o o -1 -1 -1 -1 An equation has an inverse if the operation gives the identity A Further Observation is that: Commutativity: a b = b a o o An operation is commutative if the order of the elements doesn't effect the answer Exercise 4A All Parts Exercise 4B Even Q's Does [{mod 5},+] form an Abelian Group (mod 5), addition 1) Construct a Cayley Table for (mod 5)

2) Perform 5 Tests for Abelian group

Closed

Associative

Identity

Inverse

Commutative Cayley Table [{mod5},+] Tests for Abelian 1. Closure

2. Associativity

3. Does an Identity exist

4. Does an Inverse exist

5. Commutativity 1. All results are of the same set, so it the set is closed

2. Addition with whole numbers is associative

3. There exists a value of 0 in the Cayley Table (Identity)

4. There exists a value of 0 in every row (Inverse)

5. If there is a line of symmetry down the leading diagonal of a Cayley Table then the set is commutative

All 5 are true so [{mod5},+] is an Abelian Group Exercise 4C Q2,4,5-9