Math 344 Winter 07
Group Theory Part 1: Basic definitions
and Theorems
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}
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
What is a group?
Definition: A group (G, •) is a set, G, together with an
operation, •, satisfying the following axioms:
1. (Associative)
(a•b)•c = a•(b•c), for all a, b, c G
2.
3.
(Identity)
There exists an element e G such that
= a for all a G
a•e = e•a
(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!
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.
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
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)
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
What are some other examples of groups?
Group
Not a Group
(R, +)
(R, •)
(Z, +)
(N, +)
(Q, +)
(Q, •)
(R\{0}, •)
({0}, -)
(Z, -)
(R+, •)
(R-, •)
(all polynomials, +) (all polynomials, •)
(Q \{0}, •)
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)
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
I
I
R
R2
F
FR
FR2
R
R
R2
I
FR2
F
FR
R2
R2
I
R
FR
FR2
F
F
F
FR
FR2
I
R
R2
FR
FR
FR2
F
R2
I
R
FR2
FR2
F
FR
R
R2
I
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
4-Group)
(Symmetries of a non-square
rectangle AKA the Klein
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.
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.
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. 
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.
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.
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
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.
That’s it for now!