(1) 4x = 3

The answer depends on what "things" we allow *x* to be. If
we are doing all our arithmetic using the integers then there is
no solution--there is no integer that gives 3 upon being multiplied
by 4. On the other hand if we are doing our arithmetic in **Z**/5
("Integers mod 5" as it's sometimes called)
then *x* = 2 is a solution. If we are using the more usual
rational number system **Q**, then the solution is *x* = 3/4.

We can gain insight into all such questions by considering the equation

(2) a • x = b

and then bringing up the question of solutions. Well, what objects
are *a* and *b*? To what class of objects is *x*
allowed to belong? What is the operation symbolized by the dot (•)?

Group theory is concerned with systems in which (2) always has a unique
solution. The theory does not concern itself with what *a* and
*b* actually are nor with what the operation symbolized by •
actually is. By taking this abstract approach group theory deals with
many mathematical systems at once. Group theory requires only that a
mathematical system obey a few simple rules. The theory then seeks to
find out properties common to all systems that obey these few rules.

The axioms (basic rules) for a **group** are:

**CLOSURE**: If**a**and**b**are in the group then**a • b**is also in the group.**ASSOCIATIVITY**: If**a, b**and**c**are in the group then**(a • b) • c = a • (b • c)**.**IDENTITY**: There is an element**e**of the group such that for any element**a**of the group

**a • e = e • a = a.****INVERSES**: For any element**a**of the group there is an element**a**^{-1}such that**a • a**^{-1}**= e**

and**a**^{-1}**• a = e**

That's it. Any mathematical system that obeys those four rules is a
group. The study of systems that obey these four rules is the basis
of **GROUP THEORY**

**Closure**

The first axiom of group theory is theCLOSURE: Ifaandbare in the group thena • bis also in the group.

**Associativity**

The group operation must beASSOCIATIVITY: Ifa, bandcare in the group then(a • b) • c = a • (b • c).

but

Thus **(5 - 3) - 2** does not equal **5 - (3 - 2).** Similarly
with division we can see that

but

Another example of a binary operation that is not associative is the
binary operation of averaging, which I will represent as *av*.
It gives the average of the pair of numbers that it acts upon. For
instance 4 *av* 6 = 5 and 9 *av*2 = 5 1/2. When trying to
find the usual average of three numbers we cannot simply apply the
binary *av* operation twice:

but

IDENTITY: There is an elementeof the group such that for any elementaof the group

a • e = e • a = a.

A group must have an **IDENTITY** element. This is an element with a neutral action. When the identity element is combined with any element of the
group in the group operation the result is always to give back the
same member of the group. For multiplication of real numbers the
identity element is 1; for addition of real numbers it is 0.
For 2 × 2 matrix multiplication the identity is

| 1 0 | | 0 1 |

The binary operation of *av*, given above, is an example of an
operation without an identity element. There is no number which when
averaged with any chosen number gives that chosen number back again.
True, for any chosen number there is a number that may be averaged
with it to give the original chosen number back again, namely itself.
However, there is no one number that works this way for any chosen
number. (Saying, "everybody has a mother" is very different from saying,
"someone is the mother of everybody.")

Cross product of three dimensional vectors is another example of a
binary operation that does not have an identity element. Since the
cross product of vector **A** with any other vector is either the
zero vector or a vector perpendicular to **A** there can be no
vector **E** with **A**×**E** = **A**. (otherwise
**A** would be perpendicular to itself!)

In order for an operation to satisfy the axiom forINVERSES: For any elementaof the group there is an elementa^{-1}such that

a • a^{-1}= e

anda^{-1}• a = e

Now let's go back to our original question: what is the solution of

In "solving" this equation we will assume that **a** and **b**
are elements of a group with
the group operation symbolized by **•**. We
are looking for the member of the group that **x** could be
replaced by to satisfy the equation. We'll use the group axioms to
"solve" the equation
in any group.

Using the **closure** axiom and the axiom for **inverses** we
operate on both sides of the equation
by the inverse of **a**. The inverse axiom says that
**a**^{-1}, the inverse of **a** exists and the
closure axiom says that the product of **a**^{-1} and
any other group element exists and is still in the group.

Now applying the **associative** axiom,

The axiom of **inverses** gives

Finally using the axiom of **identity** we get,

So we "solved" equation (2) without answering the questions about
what **a, b** or **x** actually were or even what the operation
indicated by **•**
was. This is the power of abstraction. Group theory is a clear
example of abstraction in modern mathematics. The emphasis is not
on the particular group we might be interested in for some given
application. The emphasis is on the basic quality of *groupness*
that all groups have in common. Once a result is demonstrated to be
valid for all groups then it is clearly valid for any specific group we
may choose.

Noticeably missing from our list of four axioms is the
**commutative** property. This is because in group theory
commutativity is not assumed. Many of the most useful and
interesting groups are not commutative.
Those groups in which
the group operation is commutative are named after the great
Norwegian mathematician Neils Henrik Abel and are called **
Abelian groups**. For Abelian groups we have the further axiom

However, this axiom cannot be used in establishing a result about groups unless we are restricting ourselves to Abelian groups.

Examples of Groups

Back to the Index

© 1998 by Arfur Dogfrey