Maths and music

Additional notes on Group Theory

Groups have many fascinating properties, and are the basis for some very interesting mathematical work. A group is a mathematical structure for which you need a set of elements, such as the symmetries of a square or other figure; the numbers on a clock face; the numbers 0 and 1; the numbers 1, 2 and 3; all the rational numbers; and so on. You also need an operation to do on your set of elements, such as the combination (usually known as composition) of symmetries; permutations of numbers; addition; multiplication; and so on.

Then you have to know that the following four axioms hold:

1. Closure: all combinations of elements give another element of the set (often shown by using a Cayley table, for a small number of elements).
2. Identity: the set contains the identity element for the operation involved.
3. Inverses: the set contains the inverse of each element (that is, the element which reverses its action).
4. Associativity: the operation is associative. This means that a(bc) = (ab)c for all a, b, and c in the set. Any operation involving addition or multiplication of numbers is associative, and can be assumed. So is the composition of symmetries.

Let's take an example and see how this works in practice. Suppose we take the numbers on the clock face, and use 'clock arithmetic', so for instance 2 + 4 = 6, but 6 + 7 = 1.

1. Closure. We can use a Cayley table to show that the set is closed (ie. that no new elements arise when any two are put together using clock arithmetic).

   +	1	2	3	4	5	6	7	8	9	10	11	12
   1	2	3	4	5	6	7	8	9	10	11	12	1
   2	3	4	5	6	7	8	9	10	11	12	1	2
   3	4	5	6	7	8	9	10	11	12	1	2	3
   4	5	6	7	8	9	10	11	12	1	2	3	4
   5	6	7	8	9	10	11	12	1	2	3	4	5
   6	7	8	9	10	11	12	1	2	3	4	5	6
   7	8	9	10	11	12	1	2	3	4	5	6	7
   8	9	10	11	12	1	2	3	4	5	6	7	8
   9	10	11	12	1	2	3	4	5	6	7	8	9
   10	11	12	1	2	3	4	5	6	7	8	9	10
   11	12	1	2	3	4	5	6	7	8	9	10	11
   12	1	2	3	4	5	6	7	8	9	10	11	12

   We can see from the Cayley table that no combination gives us anything other than a number on a clock face, and so this set with this operation is closed.

2. InverseYou can see which element is the inverse by seeing which one repeats the row and column headings, since this means it doesn't change the elements. In this case it is the number 12.
3. Inverses. You can find the inverse of each element by seeing which other element makes it equal to the inverse. An element and its inverse put together make the inverse. Inverses are listed in the table below:

Element	1	2	3	4	5	6	7	8	9	10	11	12
Inverse	11	10	9	8	7	6	5	4	3	2	1	12

12 is self-inverse, and every element has an inverse in the set.

4. Associativity. All addition is associative. To prove this, let's take three numbers x, y, and z.
   
   
   
   
   

   All four of the axioms are therefore satisfied by the integers from 1 to 12, and the operation of 'clock arithmetic', and so this is a group.

Let's look at an example now of a set of numbers which does not form a group with addition: {1, 2, 3, 4}.

1. Closure.

   +	1	2	3	4
   1	2	3	4	5
   2	3	4	5	6
   3	4	5	6	7
   4	5	6	7	8

   All the cells in this Cayley table with a red background colour show instances where closure is broken.

2. Inverses. Looking at the Cayley table, there is no element whose row and column repeats that of the row and column headings, so the inverse is not contained in the elements (it is 0 for addition).
3. Inverses. Since we don't have an identity, we can't have inverses.
4. Associativity. All addition is associative, so this is also.

Only one of the four axioms holds in this case, and that is not sufficient. So {1, 2, 3, 4} with addition is not a group.

See if you can find out if the following sets of elements and operations are groups (you can assume that associativity holds in all cases):

1. The numbers {1, -1} and addition.
2. The numbers {1, -1} and multiplication.
3. The symmetries of the equilateral triangle (there are six), and composition of symmetries.
4. The odd and even numbers, and addition.
5. The odd and even numbers, and multiplication.
6. The real numbers without zero, and multiplication (you will have to argue for closure, etc, rather than using a Cayley table).
7. The integers and addition (this will also require an argument).

Answers

To find out more about groups, you could try the following websites (you may find some of what they contain a bit difficult, but just skip over those bits, and try to get a flavour of what it is about):

http://members.tripod.com/~dogschool/
http://en.wikipedia.org/wiki/Group_(mathematics)

top of page