Great Theoretical Ideas In Computer Science Powerful Tools     ! Lecture 14 CS 15-251

Build your toolbox of abstract structures and concepts Build your toolbox of abstract structures and concepts. Know the capacities and limits of each tool.

Today’s Lecture: groups

Inverse, Closure Associativity Inverse Identity What makes this calculation possible are abstract properties of integers and addition. Closure: the sum of two integers is an integer Associativity: (x + y) + z = x + (y + z) Identity: there is an integer 0 such that x, 0 + x = x + 0 = x Inverse: x  an integer –x s.t. x + (-x) = (-x) + x = 0

closure, identity, associativity, inverse integers / addition naturals / addition odd integers / addition even integers / addition rationals / addition reals / addition complex numbers / addition all four not inverse not closure, not identity

Definition of a Group Let S be a non-empty set. Let  be a binary operator on S. (S,) is called a group if it has these properties: closure: associativity: identity: inverse:

Commutativity A group (S,) is commutative if A commutative group is also called an Abelian group.

integers / + naturals / + odd integers / + even integers / + rationals / + integers /  rationals /  rationals – {0} /  group not inverse not closure, not identity

0 1 0 0 1 1 1 0 is a group closure associativity identity: 0 inverse:

Cancellation Theorem: Proof: inverse, closure associativity inverse identity

Is this a group? Identity? 1 But 0 has no inverse!

group 1 2 1 1 2 2 2 1 group  1 2 3 1 1 2 3 2 2 0 2 3 3 2 1 Not: closure inverse

 1 2 3 4 1 1 2 3 4 2 2 4 1 3 3 3 1 4 2 4 4 3 2 1 group Notice that each row and column is a permutation of the elements.

Theorem: Each row and column of the multiplication table is a permutation of the group elements. Proof: Suppose not. By closure, if a row is not a permutation, it must have repeated elements. By cancelation:  b  c  a x x 

Conjecture: Sp is a group for prime p.  1 2 3 4 5 1 1 2 3 4 5 2 2 4 0 2 4 3 3 0 3 0 3 4 4 2 0 4 2 5 5 4 3 2 1 Not: closure inverse means that So  is not closed. Conjecture: Sp is a group for prime p.

 a-1 a 1 Theorem: Suppose (S,) has closure associativity identity cancelation And S is finite, Then (S,) is a group. Therefore, every a has an inverse, a-1, such that a  a-1 = 1. a-1 1  a Proof: This row is a permutation of the elements of S.

Cancelation modulo n Theorem: Proof: Corollary:

Theorem: with multiplication modulo n is a group. Proof: associativity. identity. closure: Let * denote multiplication modulo n. Suppose (a*b, n) > 1. Then there is a prime p such that p | n and p | a*b. p | a*b  p | ab-kn, for some k (note: ab is NOT modulo n)  p | ab (since p | n)  p | a or p | b (since p is prime)  (a,n)  p or (b,n)  p (a contradiction) cancelation:

A permutation  on [1. n] is a one-to-one function [1 A permutation  on [1..n] is a one-to-one function [1..n] mapping onto [1..n]. 1 1 2 2 3 3 4 4

Composition of permutations Let 1 and 2 be permutations on [1..n]. The composition of 1 and 2, written 1  2 is given by 1  2 (x) = 1 ( 2 (x) ) x 2 1 1  2 (x)

“2 composed with 1” Notice: Example: Composition of permutations is not always commutative.

An = set of all permutations on [1..n]  = permutation composition Theorem: (An , ) is a group Proof: closure a a-1 inverse a b c associativity identity

Subgroups Let (S,) be a group. Then H is a subgroup of S if H  S and (H,) is a group.

Example: (H,+) is a group.

Lagrange’s Theorem: If H is a subgroup of a finite group G, then the size of H divides the size of G. Example:

Proof of Lagrange’s Theorem (slide 1 of 3): Definition: Lemma: Proof: If aH were smaller it would mean but by cancelation,

Proof of Lagrange’s Theorem (slide 2 of 3): Definition: Lemma: Proof: Suppose Then Similarly

Proof of Lagrange’s Theorem (slide 3 of 3): Lemma: Proof: Every element in G appears in at least one of these. Thus, this list contains a finite list of distinct sets, all of size |H|, that partition G.

Example: