Many mathematical objects have some degree of symmetry - by this, we mean that we can do something to them which doesn't change what they look like. As an example, consider a square. We can rotate it by 90 degrees (a quarter turn), 180 degrees (a half turn) or 270 degrees (a three quarter turn) and it will look the same afterwards. We can also reflect it in its horizontal and vertical axes of symmetry, or either of its diagonal axes of symmetry, and it will look the same afterwards.
More importantly, we can perform any of these rotations or reflections in a row, and the square will still look the same. If we call any rotation or reflection of the square a transformation then we can talk of composing transformations, i.e. doing them one after the other. We say that any composition of transformations gives us another transformation.
We gave seven examples of transformations here (three rotations and four reflections). Are there any more symmetries? In fact there is just one more, which is the most obvious one of all. We could simply do nothing to the square, which will obviously leave it looking the same! This gives us eight symmetries in total. It might seem weird to think of "doing nothing" as a transformation, but it is important that we do so -- you will see why later.
As a last point in this introduction, we should point out that you can 'undo' any of the transformations. If you rotate the square, you can 'undo' that rotation by rotating the square in the opposite direction, which is the same as doing nothing. If you reflect the square in one of its axes of symmetry, you can 'undo' that reflection by reflecting again in the same axis of symmetry. This is why it's important to count doing nothing as a transformation -- we want the composition of any two transformations to be another transformation.
This all looks very interesting, and it would be good to see if we can find something like a general mathematical theory for talking about symmetry in this way. In fact we can, and that is what we call group theory.
We define a group formally by saying that a group G is a set (whose elements are our transformations) with a binary operation , which we use to talk about the composition of transformations. Formally, is a function from G G G. To be a group, our set G and its binary operation have to satisfy four axioms:
- A1. If a and b are elements of G then a b must also be an element of G. If this is true then we say that G is closed.
- A2. The operation must be associative, i.e. for any a, b, c ∈ G we must have a (b c) = (a b) c.
- A3. We must have an identity element e ∈ G, which represents doing nothing. The identity element satisfies e g = g, and g e = g.
- A4. We must be able to undo any of our transformations. We say that for every g ∈ G we must have an inverse element g-1, which satisfies g-1 g = e, and also g g-1 = e.
These four properties of closure, associativity, identity and inverse define a group.
Examples of groups
Symmetry group of the square
We already saw that the transformations of a square which leave it unchanged form a group. We can express this in a more mathematical language. Let's write ρ as a shorthand for the transformation "rotate by 90 degrees". If we want to do a 180 degree rotation then we simply do two 90 degree rotations one after the other. We write this using our composition operation, as ρ ρ, which we read as "perform ρ, then perform ρ again". In fact we can compress this notation even more -- we think of composition as something like a multiplication, and we just write ρ2 to mean ρ ρ.
Similarly, if we want to do a rotation by 270 degrees then we simply rotate by 90 degrees three times, and we write this as ρ3. Now if we rotate once more, then we are back to where we started -- that is, rotating by 360 degrees is the same as not rotating at all! If we use the notation e for our identity element "do nothing" then we can write ρ4 = e.
The reflections are a little more complicated. We will use the notation m to mean "reflect in the vertical axis of symmetry", i.e. reflecting the square left-to-right. Now we could come up with new symbols for the other three reflections, but it turns out that each of the other reflections are just combinations of rotations with the left-right reflection! It's easiest to see this with the help of a picture:
The integers Z form a group under the operation of addition. To check this, we just need to check that each of our four properties of closure, associativity, identity and inverse are satisfied. We can do these one by one:
- A1. (Closure). We need to know that whenever a, b ∈ Z then a + b ∈ Z as well. But this is obvious, because adding together two whole numbers always gives another whole number!
- A2. (Associativity). We want to show that whenever a, b, c ∈ Z we have (a + b) + c = a + (b + c). But again this is obvious, because brackets don't matter when we're adding numbers together.
- A3. (Identity). We want to find an integer which leaves every other number unchanged when you add it -- but this is exactly what zero does. We have a + 0 = a and 0 + a = a, so our identity is 0.
- A4. (Inverse). For every integer a ∈ Z we want to find an integer b such that a + b = b + a = 0. But if we choose b = -a then this is satisfied for every a, and so the inverse of a is just -a.
Since all four of our properties are satisfied, Z must be a group under addition.
Notice that Z is not a group under the operation of multiplication. The identity would have to be 1, since a x 1 = 1 x a = a. But then 0 x a = 0 for every integer a, so we could never find an integer a which is the inverse of 0. Since 0 doesn't have an inverse, Z is not a group under multiplication. This is an important point to notice -- the operation that we choose is just as important in defining the group as the set which we apply that operation to.
In our introduction, we said that groups were associated with symmetries of objects. What object are the integers the symmetry group of? The answer is curious -- the integers are the symmetry group of themselves! An intuitive picture of the integers is as points on a number line, stretching off to infinity in both directions. If we add an integer a to every number on this number line then we will just get another copy of the number line. What used to be the point -a will now be the point 0, and what used to be the point 0 will now be the point a. However, looked at as a whole, the number line will look exactly the same as it did before you added a to every number.