Group Theory
July 14, 2012
Definitions and Examples
Definition of a Group
A set S is a group under an operation ◦ if:
1. For every two elements s, t ∈ S, s ◦ t ∈ S. This is called closure under the group operation.
2. For every three elements s, t, r ∈ S, (s ◦ t) ◦ r = s ◦ (t ◦ r). This is the associative property.
3. There exists an element e ∈ S such that, for every element s ∈ S, e ◦ s = s ◦ e = s. This
element is called the group’s identity element.
4. For every element s ∈ S, there exists a (not necessarily different) element s−1 ∈ S such that
s ◦ s−1 = s−1 ◦ s = e. s−1 is called the inverse of s.
Note that commutivity is not required to have a group. However, a group G is called commutative,
or abelian. if for every s, t ∈ G, s ◦ t = t ◦ s.
To show that something is a group, you must show that the four above statements hold. To show
that something is not a group, you need show only that one of the statements does not hold.
Examples
The Reals
Theorem: The real numbers form an abelian group under addition.
Proof. We show the above properties hold.
1. The sum of any two real numbers is a real number, so the reals are closed under addition.
2. Addition on the real numbers is associative.
3. The identity element is 0, since for every t ∈ R, t + 0 = o + t = t.
4. The inverse of t is −t for all t ∈ R, since t + (−t) = (−t) + t = 0.
5. Addition of real numbers is commutative.
Theorem: The real numbers do not form a group under multiplication.
Proof. Although 3 of the 4 group properties hold, statement 4 does not. The identity under
multiplication would be 1, as t · 1 = 1 · t = t for all t ∈ R. However, 0 ∈ R and there is no t ∈ R
such that 0 · t = 1. Therefore, not every element in R has an inverse, and R is not a group under
multiplication.
Theorem: The set R\{0} forms an abelian group under multiplication.
Group Theory
July 14, 2012
Proof. We show the above properties hold.
1. The product of any two non-zero real numbers is itself a non-zero real number, so the set is
closed under multiplication.
2. Multiplication of any real numbers is associative.
3. The identity element is 1, as t · 1 = 1 · t = t for any t ∈ R\{0}.
4. The inverse of any t ∈ R\{0} is 1t , which is itself in R\{0}.
5. Multiplication of any real numbers is commutative.
The Complex Numbers
The above results also hold for the set C by the same arguments.
The Integers
The integers are also a group under addition by the same arguments above.
Theorem: Neither Z nor Z\{0} forms a group under multiplication.
Proof. The identity would again be 1, but only 1 and -1 would have inverses within the set Z or
Z\{0}. Therefore, neither set is a group.
Theorem: The positive integers, negative integers, nonpositive integers, and nonnegative integers
do not form groups under addition or multiplication.
Proof. 2 is a positive and nonnegative integer, but its additive and multiplicative inverses are
neither a positive nor a nonnegative integer. Similarly, -2 is a negative and nonpositive integer,
but its additive and multiplicative inverses are neither a negative nor a nonpositive integer.
Other Examples
These results are presented without proof and are primarily to demonstrate that other structures
can be groups.
• The set of all n by n invertible matrices form a non-abelian group under multiplication.
• The set of all rotations and reflections that maintain the symmetry of a regular n-gon is a
finite abelian group under composition. For example, there are 6 distinct symmetries of an
equilateral triangle, corresponding to:
– Doing nothing to the triangle (the identity element)
– Rotating the triangle clockwise by 120 degrees
– Rotating the triangle counterclockwise by 120 degrees
Group Theory
July 14, 2012
– Reflecting the triangle about an axis of symmetry
– Reflecting the triangle about an axis of symmetry and then rotating it clockwise by 120
degrees
– Reflecting the triangle about an axis of symmetry and then rotating it counterclockwise
by 120 degrees
Performing multiple operations will always yield the same result as an operation in the group,
it can be shown that function composition is always associative, and every operation has an
inverse that returns the triangle to its original position.
Basic Corollaries
The following give a sense of how to proceed with basic proofs involving groups and explain some
properties of the groups above.
Theorem: There is only 1 identity element in every group.
Proof. By contradiction.
Suppose that a and b are identity elements in a group G such that a 6= b. Consider a ◦ b. Since
a is an identity element in G, a ◦ b = b. However, since b is an identity element in G, a ◦ b = a.
Therefore, a = b, a contradiction, and there cannot be multiple identity elements in a group.
Theorem: If a ◦ x = a ◦ y for a, x, y ∈ G, then x = y.
Proof. Since a ∈ G, there exists some inverse b ∈ G such that b ◦ a = e. Therefore:
x = e ◦ x = (b ◦ a) ◦ x = b ◦ (a ◦ x) = b ◦ (a ◦ y) = (b ◦ a) ◦ y = e ◦ y = y
Note that this implies that the inverse of any element a is unique. If x and y were both inverses
of a, then a ◦ x = a ◦ y = e, where e is the identity element. The above theorem then implies that
x = y, a contradiction.
Theorem: (a−1 )−1 = a for every a ∈ G
Proof. Note that a−1 ◦ (a−1 )−1 = e = a−1 ◦ a. The above theorem implies that if a−1 ◦ (a−1 )−1 =
a−1 ◦ a then (a−1 )−1 = a.
Theorem: For every a, b ∈ G, (a ◦ b)−1 = b−1 ◦ a−1
Proof. Note that a ◦ (b ◦ (a ◦ b)−1 ) = (a ◦ b) ◦ (a ◦ b)−1 = e. Left-multiplying both sides by a−1 gives
a−1 ◦ (a ◦ (b ◦ (a ◦ b)−1 )) = a−1 . We can cancel the a−1 ◦ a on the left side, giving b ◦ (a ◦ b)−1 = a−1 .
We left-multiply both sides by b−1 to get (a ◦ b)−1 = b−1 ◦ a−1 .
Group Theory
July 14, 2012
Subgroups
A non-empty subset S of a group G is called a subgroup if S is itself a group.
To show that a subset S of a group G is a subgroup, it sufficies to show that:
• S is closed under the operation of G.
• If a ∈ S, then a−1 ∈ S as well.
This is sufficient because if G is a group, then associativity holds in G and in all subsets of G.
Moreover, the above conditions imply that S has an identity, since, for any a ∈ S, a−1 ∈ S and
a ◦ a−1 ∈ S.
Examples
• Every group has at least 2 subgroups - the identity element and the group itself.
• The real numbers are a subgroup of the complex numbers under addition.
• The integers are a subgroup of the real numbers (and thus of the complex numbers) under
addition.
• The set {1, −1} is a subgroup under multiplication of the group R\{0}.