When I started to study "algebra" at university, I was surprised to discover that it looked nothing like the "algebra" I had studied at school. Gone were the algebraic expressions and quadratic equations, and in came a whole new set of words and symbols.

But it was still to do with generalising. In school-level algebra, we can generalise results that work for lots of different numbers (such as (x-1)(x+1) º x2-1 , or find a formula that generalises a sequence of numbers (nth term = 3n + 4). The algebra studied at university makes connections between more disparate areas of mathematics, such as arithmetic, combinatorics and symmetry. This is very powerful; if we can show that two situations behave in the same way, then if we find something interesting about one situation, there will be an equivalent result in the other situation.


So algebraists look for ways to describe seemingly different situations in the same way. They will tend to describe them in terms of a set of elements, and one or more operations, which are ways of combining elements. This is quite difficult to understand without seeing some examples, so let's explore some:

1) Imagine taking the numbers 0, 1, 2 and 3. These are the elements. We're going to add them, but we'll do this "mod 4"; that just means that we'll write down the remainder when the answer is divided by 4. This is the operation. So, for example, 2 + 3 = 5 = 1 mod 4.
We can build up a table of the answers we get:
+ 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1
2
(this table (and the rest) would be better with a bolder line between the header row and column and the rest)
Here are some more sets of 4 elements, each time with an operation. Try to complete each table, then click here to see if you are right. Some involve arithmetic, some involve symmetry, and one involves looking very silly.
2) Take the numbers 2, 4, 6, 8 and multiply them mod 10 (so just write down the last digit).
x 2 4 6 8
2
4
6
8
3) Take the numbers 1, 2, 3, 4 and multiply them mod 5.
x 1 2 3 4
1
2
3
4
4) Take the numbers 1, 3, 5, 7 and multiply them mod 8.
x 1 3 5 7
1
3
5
7
5) Take a square; you may find it helpful to use this resource. link to Shuffles page set up with square and rotations only Our elements this time will be "the rotations of a square". We could leave it as it is(call this I), we could turn it through 90° anticlockwise (call this Q), we could turn it through 180° (call this Q²), or we could turn it through 270° anticlockwise (call this Q³). Our operation this time is just doing one element after another. So Q²Q³ is turning through 180° then 270° or do we need to work right to left? , which is the same as turning through 90°, so the answer is Q.
I Q Q² Q³
I
Q
Q²
Q³
6) Take a rectangle. Ideally you need a rectangle of clear plastic, with each corner painted a different colour, but paper will do. We're going to be interested in where the colours move to, but not which way up the plastic is. This time our elements will be the symmetries of a rectangle; the ways we could move it so that it is still in the same orientation. We could leave it as it is (I), or we could flip it vertically or horizontally (V, H), or we could rotate it through 180° (R). Again, the operation is just going to be doing one followed by another; to start you off, VH = R.
I V H R
I
V
H
R
7) Take a T-shirt, one where the front and back are clearly different. In order that you don't get too many strange looks, you might like to try this out in the privacy of your bedroom! If you take the T shirt off and put it on again, there are four things you could do in between. You could leave it as it is (Same), you could turn it back to front (BTF), you could turn it inside-out (I-O), or you could do both of these (Both). These are our four elements, and the operation is just doing one after another.
Same BTF I-O Both
Same
BTF
I-O
Both

Now have a good look at the tables you have completed, and look at the similarities and differences.
All these tables have a number of things in common:
Mathematicians call this structure a group. Not all groups have four elements (they could even have an infinite number), but they all have tables which share most of the properties above.
Put more formally, a group is a set of elements and an operation which have the following properties, where a, b, etc are elements, and * is the operation:
The colours show how these properties link to the properties of the tables observed above.
The symmetry of the tables is not part of the definition of a group. However mathematicians define an Abelian group as one where for any pair of elements a and b, a*b=b*a; all the examples above are Abelian.

Look at the groups above and identify which element is the identity element in each group. You could also identify the inverse of each element. Some elements are self-inverse ; a*a=e.

We've looked at the properties shared by the tables above. Now look again at the tables. How many different tables are there?


You probably thought there were three. However, try filling in these two again:
2) (multiplication mod 10)
x 6 8 2 4
6
8
2
4
3) (multiplication mod 5)
x 1 3 2 4
1
3
2
4

Rearranging the elements shows you that all of these examples fit one of just two structures.

We have seen that the tables are not reliable as a way of distinguishing; one useful way to start to describe individual groups is to look at the order of the elements in the group. The order of an element is the number of times it needs to be combined with itself to get the identity element. For instance, if we turn te T-shirt inside-out twice, we're back where we started, so the order of this element is 2. If we rotate a square through 90°, we have to do it 4 times, whereas if we rotate it through 180°, the order is just 2. The identity element obviously has order 1, and elements which are self-inverse have order 2.

There are in fact only two different groups of order 4 (consisting of 4 elements).
K4 (Klein-4) is the name given to a group with 4 elements where all elements other than the identity are self-inverse.
C4 is the name given to a cyclic group of 4 elements. A cyclic group is one where one (or more) elements are of the same order as the group; all the other elements are created by combining this element with itself.
With a bit of thinking, you may be able to convince yourself that there are no other groups of order 4.

This article has explored just one type of mathematical structure, and indeed has only looked at groups of order 4. In fact, groups can have any number of elements, and much of the more interesting work involves those with an infinite number of elements. For instance, the integers form a group under addition (check those properties), but not under multiplication, as the elements do not have inverse elements which are in the group. Other mathematical structures include rings and fields . There are also various extra classifications; we've met the term Abelian group, for example. You can get an idea of the variety of the language used by looking here, but don't expect to understand all the definitions at this point!


Further exploration

We've looked at a couple of symmetry groups; one was the rotations of a square, and the other was the symmetries (both reflections and rotations) of a rectangle.
Using the Shuffles resource, you can explore the symmetries of various regular polygons. For example, the set of symmetries of an equilateral triangle, the set of rotations of a regular hexagon and the set of reflections of a regular hexagon all have six elements. Are they groups (check the properties)? Are they the same group?

Are there any elements of order 3 in the groups explored in this article? Can you explain? What about groups with different numbers of elements? What orders are the elements in these groups?

There are various problems involving groups in the March 2005 issue of the site.