Group Theory: Closure, Identity, Inverses, and Cayley Tables, Study notes of Mathematics

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Basic Group Theory I

1. What is GROUP THEORY?

1. What is GROUP THEORY?

We can gain insight into all such questions by

considering the equation

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 (•)?

1. What is GROUP THEORY?

Group theory is concerned with systems in which “ a • x = b ” 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 a few rules.

CLOSURE

If a and b are in the group then a • b is also in the group.

The first axiom of group theory is the CLOSURE axiom. For a system to be a group the binary operation (symbolized here by "•") must be valid for any pair of elements in the group and the result of the operation must be an element of the group. The set of negative integers, for example, is not closed under multiplication because the product of two negative integers is not a negative integer.

CLOSURE

The set of vectors of unit magnitude is not closed under vector addition because the vector sum of two unit vectors is not necessarily a unit vector. More interestingly, the set of three-dimensional vectors is not closed under scalar (sometimes called "dot") product since the scalar product of two vectors is not a vector. On the other hand the counting numbers are closed under addition and multiplication. They are not closed under either subtraction or division. The set of three dimensional vectors is closed under vector (or "cross") product since the result of the operation is a three-dimensional vector.

ASSOCIATIVITY

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:

2 av (3 av 7) = 2 av 5 = 3 1/ and (2 av 3) av 7 = (2 1/2) av 7 = 4 3/

IDENTITY

There is an element e of the group such

that for any element a of the group

a • e = e • a = a.

IDENTITY

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.")

IDENTITY

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!)

INVERSES

In order for an operation to satisfy the axiom for INVERSES the operation must have an identity element. Thus we know without further investigation that the av operation and the cross product operation do not have inverses since they do not have identity elements. An inverse element is a way to undo an operation. For example suppose I add 7 to a number and get the result 12. If I want to undo the addition of 7 and return to my original number that I added 7 to I simply add the inverse of 7 which is -7. This shows that my original number that I added to 7 was 5. I undid the addition. Similarly I can undo multiplication by 2 by multiplying by the inverse of 2 which is 1/2.

INVERSES

Many of the extensions of the number systems of arithmetic were made to create inverses. The integers are an extension of the natural numbers in which addition has an inverse. The rational numbers are an extension of the integers in which each non- zero number has an inverse under

multiplication. A 2 × 2 matrix may or may not

have an inverse under matrix multiplication. Those matrices which do not have multiplicative inverses are called singular.

1. What is GROUP THEORY?

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.

a-1^ • (a • x) = a-1^ • b

1. What is GROUP THEORY?

Now applying the associative axiom,

(a-1^ • a) • x = a-1 • b

The axiom of inverses gives

e • x = a-1^ • b

Finally using the axiom of identity we get,

x = a-1^ • b