Some groups are infinite. The group of integers under addition is an infinite group. There are an infinite number of integers so the addition group of integers has an infinite number of elements. Some groups are finite. The Attayun-HOOT! group given in the examples lesson is a finite group with four elements.
In elementary school we were able to show the multiplication table for the one digit numbers in a square array:
× | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
2 | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
3 | 0 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 |
4 | 0 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 |
5 | 0 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 |
6 | 0 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 |
7 | 0 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 |
8 | 0 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 |
9 | 0 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 |
we find the product 5 × 7 by finding 5 in the left hand edge, finding 7 along the top edge. We then find their product, 35, in the intersection of the 5 row and the 7 column. Similarly we can find the product 7 × 5 in the intersection of the 7 row and the 5 column. The fact that multiplication of numbers is commutative is readily visible by noting that the multiplication table is symmetrical about the main diagonal--the diagonal with the squares on it, 0, 1, 4, 9 . . . etc. The main diagonal is the one from the upper left corner to the lower right corner.
Similarly the binary operation for a group can be represented in a square array just as the multiplication table was. For example here is the "multiplication" table for the Atayun-HOOT! group. (We've let A stand for Attention, R stands for Right face, L stands for Left face and B stands for aBout face.)
• | A | R | L | B |
---|---|---|---|---|
A | A | R | L | B |
R | R | B | A | L |
L | L | A | B | R |
B | B | L | R | A |
To find the group element that corresponds to L • R we find the intersection of the L row and the R column. There we note that L • R = A
Again by noticing the symmetry about the main diagonal we can see that this is an abelian group. We can also see that A is the identity as the A-row reproduces the top row and the A-column reproduces the left column.
Notice that A, the identity element appears exactly once in each row and exactly once in each column (we're not counting the topmost row nor the leftmost column; they're the "input" space). This means that each element has an inverse. For each element there is something that combines with it according to the group operation to produce the identity. Notice further that the A's are distributed symmetrically about the main diagonal. This means that each right inverse is also a left inverse. In any Cayley table of a finite group the identity elements must be distributed symmetrically about the main diagonal. This follows because the axioms require each left inverse to be a right inverse.
Also notice that each row and each column of the inner table contains each element exactly once. This reflects the cancellation law. If, for instance, element B appeared twice in, for example, the L row (say once in the "C" column and once in the "D" column) it would mean that
or
Now applying the cancellation law we get C = D. But C is not equal to D. This violation of the left cancellation law which we know must be valid for all groups tells us that any Cayley table for a group can't have the same element twice in the same row. Similarly the right cancellation law tells us that the Cayley table for a group cannot have the same element twice in the same column.
Consider the following Cayley table. Does it represent the Cayley table of some group?
• | A | B | C | D | E |
---|---|---|---|---|---|
A | A | B | C | D | E |
B | B | A | D | E | C |
C | C | D | E | A | B |
D | D | E | B | C | A |
E | E | C | A | B | D |
Well, in the inner table each element appears exactly once in each column and exactly once in each row. It passes that test. It has an identity element, namely A, so it passes that test. Since the identity element appears in each column and in each row we can conclude that each element has a right and a left inverse. So far it's looking like a group. However the identity element, A is not distributed symmetrically about the main diagonal. This means that a left inverse is not necessarily a right inverse. For example, in this "not quite a group" the Cayley table tells us that C is a left inverse of D since
where, remember, A is the identity. But C is not a right inverse of D since
and B is not the identity. Thus this cannot be the Cayley table for a group as in a group every right inverse is also a left inverse.
The one group property that is difficult to check visually from a Cayley table is the associative property of the binary operation.. The associative property states that for any x, y and z in the set it is true that
Now in a system with 5 elements there are 5 choices to replace x and 5 choices to replace y and 5 choices to replace z. This gives 5 × 5 × 5 = 125 combinations that have to be checked to find out if the associative law holds! This would be a lot of work. For groups with more than 5 elements there would be an even larger number of combinations. (a system with 10 elements would have a thousand combinations to check)
In deciding whether a system is a group or not we usually will use some means other than brute force checking of the Cayley table to establish the associative property.
Just because two Cayley tables look different does not mean that they are tables of different groups. Look at the following table and compare it to the table we presented for the Atayun-HOOT group:
• | A | B | R | L |
---|---|---|---|---|
A | A | B | R | L |
B | B | A | L | R |
R | R | L | B | A |
L | L | R | A | B |
Even though the elements of the group, A, L, R and B are not distributed the same in the table it still represents the same Atayun-HOOT! group. The columns and rows have only been given in a different order. It is still true that B • L = R (and the same for all other products) no matter which table you use. It is not simply a matter of glancing at the Cayley table of a group to notice whether it is the same group as is represented in a different Cayley table. It must happen that all products represented are the same.
Furthermore, the notation we used was rather arbitrary. We could have let E stand for "Attention!" and let A stand for "About face!." Then the Cayley table could look like this:
• | E | A | R | L |
---|---|---|---|---|
E | E | A | R | L |
A | A | E | L | R |
R | R | L | A | E |
L | L | R | E | A |
This is still the table for the Atayun-HOOT group.
Let's look at yet another group. The group of the real and imaginary units under multiplication. This group consists of {1, -1, i, -i} and the group binary operation is usual multiplication. The Cayley table for this group is:
× | 1 | -1 | i | -i |
---|---|---|---|---|
1 | 1 | -1 | i | -i |
-1 | -1 | 1 | -i | i |
i | i | -i | -1 | 1 |
-i | -i | i | 1 | -1 |
However if we re-symbolize the Atayun-HOOT! so that
These two groups, one dealing with military marching orders and the other dealing with multiplication of real and imaginary numbers can be represented by the same Cayley table. They are somehow, underneath all the glitter, the same group. This is an example of the concept of isomorphism. The two groups have exactly the same shape (iso -- same; morphism -- shape). Thus anything we know about the basic mathematical structure of the Atayun-HOOT group will be true about the {1, -1, i, -i} group.
In abstract algebra (of which group theory is a main part) we are only interested in the basic mathematical structure of groups. We don't study the value of marching in building morale among the troops and engendering group identity (no pun intended). This abstraction is a powerful feature of modern mathematics. Once the basic mathematical pattern underlying something is understood then we suddenly know a lot about any other system that also has that same basic underlying mathematical pattern. Thus we already know that for any system which satisfys the group axioms there is a unique identity, each element has a unique inverse and the cancellation laws hold.
Bertrand Russell summed up this modern, abstract view in his famous epigram, "Mathematics may be defined as the subject where we never know what we are talking about, nor whether what we are saying is true."
© 1998 by Arfur Dogfrey