Says Roger:

Group theory is the study of one of the most basic structures
of modern mathematics, the group. The definition of a group
will be developed via numerous examples from the integers and
rational numbers. Some examples of groups of symmetries of
regular polygons (e.g., equilateral triangles) will be presented.
Lastly, Lagrange's theorem, the fundamental theorem of group
theory, will be presented and proved. (Don't worry, even your
grandmother can follow this proof.)

Present were Jeff, Kevin, Glenn, Meredyth, Roberto, Don, Curtis, Rachel, Marvin and Roger.

Glenn's hospitality and his freshly baked banana bread have become a tradition. Together with the sumptuous provisions brought by the participants this makes every discussion evening a culinary adventure. What follows are Roger's notes which he distributed to us all at the meeting. They convey the substance but not the charm, the wit and the humor in Roger's enthusiastic performance.

Notes for a One-Lecture Introduction to Group Theory

Group theory is the study of one of the most basic structures of modern mathematics, the *group*. In this lecture the definition of a group is developed via examples from the integers and rational numbers. Also the group of symmetries of the equilateral triangle is shown. Lastly, Lagrange's theorem, the fundamental theorem of group theory, is presented and proved.

Some Number Systems Used as Sources of Examples

Binary Operators

Let S be any set. Then, a *binary operator* on S is a function which takes as input two elements of S and produces as output one element of S.

For example, let S = **N**, the set of natural numbers, and let the operator be addition of natural numbers, denoted by +. Given any two natural numbers, say b and c, then b + c is also a natural number. Thus, + is a binary operator on the natural numbers.

Questions:

Is multiplication of natural numbers a binary operation?Is subtraction of natural numbers a binary operation?

Is division of natural numbers a binary operation?

We say that a set S is *closed* under an operation if the output from the operation is always a member of S. Thus, the natural numbers are closed under addition and under multiplication, but the natural numbers are not closed under subtraction and division. In general, a set S is closed under a binary operator on S.

The Integers

That the natural numbers are not closed under subtraction is one of the reasons why the *integers* were invented. The integers consist of the natural numbers, zero, and the negatives of the natural numbers.

Thus, 3 - 5 is meaningful in the world of the integers. In fact, 3 - 5 = -2. So if you have $3 in the bank and you write a check for $5, then you have -$2 left in your account. [Most banks interpret this negative balance as an overdraft, i.e., you owe them.]

Questions:

Are the integers closed under addition?Are the integers closed under multiplication?

Are the integers closed under subtraction?

Are the integers closed under division?

Associative Operators

Let * denote any binary operator on a set S. Then, if

(a * b) * c = a * (b * c)

for every a,b,c in S, we say that *
is *associative*.

Questions:

Is addition of natural numbers associative?

Is multiplication of natural numbers associative?

Is addition of integers associative?

Is multiplication of integers associative?

Is subtraction of integers associative?

Is division of integers associative?

Identity Elements

Let * denote a binary operator on a set S. Then, if there is an element e in S with the property that

b * e = b and e * b = b,

then e is called an *identity* element for the operation *
.

Questions:

What are the identity elements for multiplication of integers?

What are the identity elements for addition of integers?

What are the identity elements for multiplication of natural numbers?

What are the identity elements for addition of natural numbers?

Inverse Elements

Let * denote a binary operator on a set S, and let e be an identity for * . Given an element b in S, if there exists an element b' in S with the property that

b * b' = e and b' * b = e,

then b' is called an *inverse* of b with respect to the operation *
.

Questions:

What are the inverses of 4 under addition of integers?

What are the inverses of 1 under addition of integers?

What are the inverses of -1 under addition of integers?

What are the inverses of 1 under multiplication of integers?

What are the inverses of -1 under multiplication of integers?

What are the inverses of 4 under multiplication of integers?

The Definition of Group

A *group* <G,*
> is a nonempty set of elements G together with a binary operator *
defined on G such that the following hold:

1. *
is *associative*, i.e., (a *
b) *
c = a *
(b *
c) for all elements a,b,c in G.

2. There exists a distinguished element e in G, called the *identity*, with the property that e *
b = b *
e = b for every element b in G.

3. For every element b in G there exists an element b' in G, called the *inverse* of b, with the property that b *
b' = b' *
b = e.

An Example of a Group

The integers under addition, denoted <**Z**,+>, is a group.

Exercise:

Prove that the statement above is true.

Questions:

Are the natural numbers under addition a group?

Are the natural numbers under multiplication a group?

Are the integers under multiplication a group?

The Rational Numbers

That the integers are not closed under division is one of the reasons why the *rational numbers* were invented. The rational numbers consist of all numbers which can be expressed as the ratio n/d of two integers n and d, where d is not zero. So, the set of rational numbers contains all fractions.

In the world of rational numbers, 3 / 5 is meaningful. It is the rational number 0.6, which we can also write as the fraction 3/5. So if you have 3 pies which you want to divide evenly among 5 people, then each person will receive 3/5 of a pie.

Questions:

Are the rational numbers closed under addition?

Are the rational numbers closed under multiplication?

Are the rational numbers closed under subtraction?

Are the rational numbers closed under division?

Another Example of a Group

The nonzero rational numbers under multiplication, denoted <**Q _{0}**, x
>, is a group.

Exercise:

Prove that the statement above is true.

Questions:

Are the rational numbers under addition a group?

Basic Facts About Groups

1. There is exactly one identity element in a group.

2. Each group element has exactly one inverse.

3. (Cancellation law) If a, b, c are group elements and a * b = a * c, then b = c.

Finite Groups and Infinite Groups

The two groups <**Z**,+> and <**Q _{0}**, x> described above each contain an infinite number of elements. However, not all groups are infinite; some contain a finite number of elements. In fact, given any natural number n, there exists a group containing exactly n elements.

Some terminology: The *order* of a group is the number of elements in the group.

A Finite Group Whose Members Are Not Numbers

We are now going to describe a finite group, the group of symmetries of the equilateral triangle. This group is not only finite, but its elements are not numbers.

We begin with an equilateral triangle whose vertices are labeled A, B, and C, and we look for all possible ways to orient the triangle so that its position in the plane does not change, but the labeling of the vertices does change:

The three-letter codes ABC, CAB, etc., identify the orientations of the triangle; note that these codes always start with the top vertex, then the left one, then the right one.

The symbols R_{0}, F_{2}, etc., stand for rotations and flips of the triangle which move it from the ABC position to the CAB position, the BAC postion, etc. Here are the definitions of these transformations:

R_{0} - Rotate the triangle counterclockwise by 0 degrees.

R_{1} - Rotate the triangle counterclockwise by 120 degrees.

R_{2} - Rotate the triangle counterclockwise by 240 degrees.

F_{0} - Flip the triangle through the axis defined by the top vertex.

F_{1} - Flip the triangle through the axis defined by the left vertex.

F_{2} - Flip the triangle through the axis defined by the right vertex.

Note that when each of the above transformations is applied to the ABC triangle, the resulting triangle is the one associated with the transformation in the diagram above. For example, when F_{2} is applied to ABC, the result is BAC, the triangle associated with F_{2} in the diagram.

We are now going to define a group named D_{6} whose elements are the six transformations listed above. The group operation is composition, meaning that b *
c is the transformation resulting when first b is applied, and then c is applied. For example, R_{1} *
F_{2} = F_{0}, i.e., a rotation by 120 degrees followed by a flip through the axis defined by the right vertex is the same as a flip through the axis defined by the top vertex.

Read the last paragraph again. Note carefully what the group elements are. They are not numbers. Rather they are functions applied to the equilateral triangle which change its orientation. The group operation for combining two functions to get another one is simply to apply the first function, then the second one. To make clear that we are in a different world than we have been up to now, note that R_{1} *
F_{2} = F_{0} and F_{2} * R_{1} = F_{1}, so

R_{1} *
F_{2} is not equal to
F_{2} *
R_{1.}

This is something that we don't encounter with addition and multiplication of integers and rational numbers: The group operation in this case is *noncommutative*.

By patiently applying one transformation after the other, we can construct a "multiplication table" for the group operation:

R |
R |
R |
F |
F |
F |

R |
R |
R |
F |
F |
F |

R |
R |
R |
F |
F |
F |

F |
F |
F |
R |
R |
R |

F |
F |
F |
R |
R |
R |

F |
F |
F |
R |
R |
R |

To determine what b *
c is, pick the row beginning with b and pick the column beginning with c, then the transformation in the cell determined by the selected row and the selected column is b *
c. For example, R_{1} starts the second row, and F_{2} starts the sixth column, so R_{1} *
F_{2} is found in the cell which is in the second row and the sixth column, namely F_{0}.

Some terminology: A table like that shown above is often called a *Cayley table*, in honor of a 19th century English mathematician who made many contributions to group theory in the early days of the subject.

Questions:

What is the identity element of the group D_{6}?

What is the inverse element of each of the elements of D_{6}?

Subgroups

Let's look at a subset of the group D_{6}, namely, the subset C_{3} = {R_{0}, R_{1}, R_{2}}. What can we say about C_{3} other than that it is a subset of D_{6}? Looking at the upper lefthand corner of the Cayley table of D_{6} above, we see that any rotation followed by another rotation is a rotation. In other words, the subset C_{3} is closed under the group operation. In fact, C_{3} is itself a group. It is a subgroup of D_{6}.

Questions:

Does D_{6} have any subgroups other than C_{3}?

Does the group <**Q**,+> have any subgroups?

Lagrange's Theorem

Note that the order of D_{6} is 6, the order of C_{3} is 3, and 6 is a multiple of 3. Note also that the order of the subgroup {R_{0}, F_{0}} is 2, and 6 is a multiple of 2. Could it be that the order of a finite group is always a multiple of the orders of its subgroups? Amazingly enough, the answer is, "Yes." This is *the* fundamental fact about finite groups.

Theorem. (Lagrange) If G is a finite group and H is a subgroup of G, then the order of G is an exact multiple of the order of H.

Proof. Let n be the number of elements in G, and let m be the number of elements in H. Let h_{1}, h_{2}, ..., h_{m} be the elements of H. Let G_{1} be the subset of G consisting of the n-m elements of G which are not in H. Thus, we have partitioned G into two disjoint subsets:

subset |
elements |

H |
h |

G |
the other n-m elements of G |

Is it possible that either of these sets is empty, i.e., can n = 0 or n-m = 0? The set H is a subgroup of G, and by definition, a group is nonempty, so H is not empty. However, if m = n, then G_{1} is empty. Is this possible? Certainly. A subgroup H may very well be the entire group G. The theorem is true in this case, because if m = n then clearly m divides n exactly (namely, once). So, we need only consider the case when m < n, i.e., when H is a proper subset of G.

Pick any g_{1} in G_{1}, i.e., any g_{1} which is an element of G but not an element of H. We know that such elements exist because G_{1} is not empty. Now multiply every element of H by g_{1}, to get the m elements g_{1}h_{1}, g_{1}h_{2}, ..., g_{1}h_{m}.

We claim that no two of the elements g_{1}h_{1}, g_{1}h_{2}, ..., g_{1}h_{m} are equal. To see this, we assume the contrary, i.e., say that g_{1}h_{i} = g_{1}h_{j} for some i and j such that i and j are not equal. Then, by the cancellation law, h_{i} = h_{j} . Since the elements of H are all distinct, it must be that i = j. This contradiction shows that no two of the elements are equal.

Let H_{1} be the subset of G consisting of the m elements g_{1}h_{1}, g_{1}h_{2}, ..., g_{1}h_{m}. We claim that no element of H_{1} is an element of H. To see this we assume the opposite, namely that some element g_{1}h_{i} of H_{1} is also an element h_{j} of H, i.e., that g_{1}h_{i} = h_{j} for some i and j. Let h_{i}' be the inverse of h_{i}, and multiply both sides of g_{1}h_{i} = h_{j} by h_{i}' to get

(g_{1}h_{i})h_{i}' = h_{j}h_{i}'

g_{1}(h_{i}h_{i}') = h_{j}h_{i}'

g_{1}e = h_{j}h_{i}'

g_{1} = h_{j}h_{i}'

Since h_{i }is an element of H, its inverse h_{i}' is also an element of H, and since H is closed under the group operation, the product of h_{j} and h_{i}' is an element of H. So g_{1} is an element of H. But we chose g_{1} so that it was not an element of H. This contradiction means that our assumption that g_{1}h_{i} = h_{j} for some i and j is false. In other words, no element of H_{1} is also an element of H.

Let G_{2} be the subset of G consisting of the n-2m elements of G which are not in H and not in H_{1}. Thus, we have partitioned G into 3 disjoint subsets:

subset |
elements |

H |
h |

H |
g |

G |
the other n-2m elements of G |

Is it possible that any of these sets is empty? We know that H is not empty, and we just proved that H_{1} consists of m distinct elements, so it is not empty. However, it is possible that G_{2} is empty. If it is empty, then n-2m = 0, so n = 2m. Thus n is a multiple of m, and the theorem is true.

If G_{2} is not empty, let g_{2} be any element in G_{2}. Multiply every element of H by g_{2} to get g_{2}h_{1}, g_{2}h_{2}, ..., g_{2}h_{m}. It is clear that by arguments similar to those used above, these m elements are distinct and none of them is an element of H. Yet more is true: None of these elements in an element of H_{1}. To see this, assume the contrary, i.e., assume that for some i and for some j we have g_{1}h_{i} = g_{2}h_{j}. Then, if h_{j}' is the inverse of h_{j}, we get

g_{1}h_{i}h_{j}' = g_{2}h_{j}h_{j}'

g_{1}h_{i}h_{j}' = g_{2}

Since h_{i}h_{j}' is an element of H, it must be that g_{1}h_{i}h_{j}' is an element of H_{1}, so g_{2} is an element of H_{1}. But this is a contradiction, because we chose g_{2} so that it was not an element of H_{1}. Therefore, if we define H_{2} to be the set consisting of g_{2}h_{1}, g_{2}h_{2}, ..., g_{2}h_{m}, then H_{2} consists of m distinct elements of G, none of which are in H or in H_{1}.

Let G_{3} be the subset of G consisting of the n-3m elements of G which are not in H and not in H_{1} and not in H_{2}. Then, we have partitioned G into 4 disjoint subsets:

subset |
elements |

H |
h |

H |
g |

H |
g |

G |
the other n-3m elements of G |

If G_{3} is empty, then n-3m = 0, so n = 3m, and therefore n is a multiple of m, proving the theorem.

If G_{3} is not empty, then proceed as above. Clearly we will eventually arrive at a situation where we have disjoint sets H, H_{1}, H_{2}, ..., H_{k-1}, G_{k}, with G_{k} empty and the other subsets each containing m elements of G. Then, we have partitioned G into k disjoint subsets:

subset |
elements |

H |
h |

H |
g |

H |
g |

... |
... |

H |
g |

G |
empty |

The set G_{k} contains n-km elements. But G_{k} is empty, so n-km = 0, or n = km. Thus, the order of G is an exact multiple of the order of H.

<End Notes>

August 1997

Roger House

e-mail: rhouse@sonic.net

End of Roger's notes and of my report.

Some of my own offerings on group theory are:

Theorem of Lagrange

Mathematical Structure of Identity

Physics As Symmetry

The Isomorphic Groups C_{3}, S_{3} and C_{3V}

August 1997

Marvin Chester

email: chester@physics.ucla.edu

9'05-