1. Math 344 Winter 07 Group Theory Part 1: Basic definitions and Theorems

2. What is an operation?
Definition 1: Let S be a set. An operation, •, on S is a function from SS to S.
• : SS  S
(Informally, an operation takes in two elements of S and outputs another element of S.)
Definition 0: SS = {(a, b) : a, b  S}

3. In the case of the symmetries of an equilateral triangle:
The operation was given by the process of combining pairs of symmetries.
e.g. R combined with F gives FR2
We used multiplicative notation for our operation so we wrote this as RF = FR2

4. What is a group?
Definition: A group (G, •) is a set, G, together with an operation, •, satisfying the following axioms:
(Associative)
(a•b)•c = a•(b•c), for all a, b, c G
(Identity)
There exists an element e G such that a•e = e•a = a for all a G
(Inverses)
For each a G there is an element a-1  G such that a•a-1 = a-1•a = e.
What about closure? We defined operations so that they are always closed!

5. Example: The symmetries of an equilateral triangle form a group with the operation of combining pairs of symmetries
(Associative) because (a•b)•c and a•(b•c) describe the exact same transformation of the triangle.
(Identity) the “do nothing” transformation is the identity since it has no effect when combined with another symmetry.
(Inverses) I, F, FR, and FR2 are all their own inverses, R is the inverse of R2, and R2 is the inverse of R.

6. More about groups of symmetries
Formally, symmetries are functions from R2 to R2. And the operation of combining symmetries is actually function composition.
So we will say that the set of symmetries of a figure form a group under composition.
Take a minute to convince yourself that the set of symmetries of any figure must form a group under composition. Remember check in this order 1) closure 2) identity 3)inverses 4)associativity

7. Note 1: Although a group (G, •) consists of both a set and an operation, when the operation is understood we often refer to the group using only the name of the set. “The group G”
Note 2: When the operation is understood, we often leave out the operation symbol and express it using multiplicative notation and denote the product simply by writing the elements next to each other (e.g. RF)

8. Abelian (commutative) groups
A group is called abelian or commutative if a•b = b•a for all a, b G.
We have seen two examples so far of commutative groups:
Symmetries of a non-square rectangle
Rotational symmetries of a square
And we have seen two examples of non-commutative groups:
Symmetries of an equilateral triangle
Symmetries of a square

9. What are some other examples of groups?
Not a Group
(R, +)
(R, •)
(Z, +)
(N, +)
(Q, +)
(Q, •)
(R\{0}, •)
({0}, -)
(Z, -)
(R+, •)
(R-, •)
(all polynomials, +)
(all polynomials, •)
(Q \{0}, •)

10. Different ways to describe groups
If the set and operation are well-known, we can just write the groups as an ordered pair with no further explanation:
Examples:
(Z, +)
(R, +)
(Q, +)
(R\{0}, •)
(R+, •)
(Symmetries of equilateral triangle, composition)

11. Different ways to describe groups (Cont)
In the case of a finite group, we sometimes just list out the elements and then use a table to define the operation.
Example: Let D6 = {I, R, R2, F, FR, FR2} with the operation given by: I R R2 F FR
FR2

12. Different ways to describe groups (cont)
Groups can also be described using generators & relations.
Example: D6 =  R, F | R3 = F2 = I, RF = FR2
The elements R and F generate the group because each element can be expressed as a product involving these two elements.
Together with the group axioms, the relations R3 = F2 = I and RF = FR2 completely determine how the operation works.
Describe the group of rotational symmetries of a square and the group of symmetries of a non-square rectangle using generators and relations.
C4 =  R | R4 = I  (Rotations of a square)
V4 =  R, F | R2 = F2 = I , RF = FR (Symmetries of a non-square rectangle AKA the Klein 4-Group)

13. The Cancellation Law Theorem 1: Let (G, •) be a group.
If a, b, c  G, then a•b = a•c  b = c and b•a = c•a  b = c .
Proof: Let e be the identity of G. Then
a•b = a•c
 a-1•(a•b) = a-1•(a•c) [Existence of inverses]
 (a-1•a)•b = (a-1•a) •c [Associative]
 e•b = e•c [Def. of inverse]
 b = c [Def. of identity]
 a•b = a•c  b = c. The proof of the second part is virtually identical.

14. Uniqueness of Inverses
Corollary 1: Each element of a group has a unique inverse.
Proof: Let G be a group and aG. Let e be the identity of G. Suppose b, c G are both inverses of a. Then ab = e and ac = e. So ab = ac. Then by cancellation law, b = c. Thus the inverse of a is unique.

15. Existence and Uniqueness of a solution to ax = b.
Theorem 2: Let a, b G. The equation ax = b has a unique solution in G.
Proof: Let e be the identity of G. Solving the equation:
ax = b
a-1(ax) = a-1b [Existence of inverses]
(a-1a)x = a-1b [Associative Law]
ex = a-1b [Def. of inverse]
x = a-1b [Def. of identity]
IF there is a solution, it must be a-1b. So we have shown uniqueness.
Note that a-1b G due to existence of inverses and closure.
Now, note that a(a-1b) = (aa-1)b = eb = b. This proves that a solution exists. 

16. The “Sudoku Property” of Group Tables
Corollary 2: Each element of a group appears exactly once in each row of the group’s operation table.
Proof: Let G be a group and a, b G. Then b appears in row a of the operation table if and only if there exists xG such that ax = b. (It would then appear in column x of row a.)
By the previous theorem, there is exactly one x in G for which ax = b. Thus b appears in exactly one column in row a.

17. It works for columns as well!
We can add a second part to Theorem 2. Equations of the form xa = b also have unique solutions. This means that each element of a group must also appear exactly once in each column of the operation table.

18. Order of a group
Definition: The order of a group G, denoted |G| is the cardinality of the set G. In the case of a finite group, the order is simply the number of elements in G.
Example: |D6| = 6

19. Order of an element
Definition: Let G be a group and aG. The order of the element a, denoted |a|, is the smallest positive integer n such that an = e (where e is the identity of G.)
Note: For an additive group, the equation changes to na = e
If no such n exists, |a| = .
Example: In D6,
|R| = 3
|F| = 2
|I| = 1
Find the order of each element in (Z, +) and (R\{0}, •)
For (Z, +), |0| = 1 and |n| = , if n  0.
For (R\{0}, •), |1| = 1, |-1| = 2 and |x| =  if x  1, -1.

20. That’s it for now!