1. Great Theoretical Ideas In Computer Science
Powerful Tools
    !
Lecture 14 CS

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

3. Today’s Lecture:
groups

4. 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

5. 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

6. 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:

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

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

9. 0 1
is a group
closure
associativity
identity: 0
inverse:

10. Cancellation Theorem:
Proof:
inverse, closure
associativity
inverse
identity

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

12. group
1 2
group 
Not:
closure
inverse

13. 
group
Notice that each row and column is a permutation of the elements.

14. 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 

15. Conjecture: Sp is a group for prime p.
Not:
closure
inverse
means that
So  is not closed.
Conjecture: Sp is a group for prime p.

16.  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.

17. Cancelation modulo n
Theorem:
Proof:
Corollary:

19. 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:

20. 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].

21. 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)

22. “2 composed with 1” Notice: Example:
Composition of permutations is not always commutative.

23. 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

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

25. Example:
(H,+) is a group.

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

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

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

29. 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.

30. Example: