The Abel Prize in mathematics, one of the awards whose laureates join us each year for the HLF, was originally proposed in 1899 as a complement to the Nobel prizes – to recognise achievement in mathematics. But what does the name Abel mean to mathematicians?
Most of Abel’s work in mathematics was done within a 6-7 year period, and covered topics including polynomial equations, elliptic functions and calculus. Many concepts in mathematics are named after Abel (so many, in fact, that Wikipedia has a dedicated List of things Named after Abel).
One of the first things on this list that I encountered when studying maths at uni was the notion of an abelian group – and to explain what one is, I’ll need to start by defining one of the most useful abstract ideas in pure maths: the theory of groups.
You’re probably very used to working with groups, but maybe don’t realise it yet. A group is a set of objects – they could be numbers, letters, permutations (as we met in my previous blog post), matrices, symmetries, or even moves on a Rubik’s cube. The main thing we need for them to be a group is this: if we take two things from in the group, and combine them using a specified rule, the result of doing that should also be something in the group.
For example, if we take two whole numbers, like 2 and 5, and add them together, the result will also be a whole number: 7. We could then take 7 and combine it with some other number, like 1, and we’d still get another number. This property is called being closed – a group, considered with the specific operation you have chosen to be the way of combining things in the group, will only ever give you things from within the group.
The definition of the operation is also in some sense part of the definition of the group. For example, I could take the same base set as in the previous example (whole numbers), and instead of adding them together, I could use multiplication. When writing this, we can use the notation (ℤ, +) to mean the whole numbers under addition, and (ℤ, ×) to mean the whole numbers under multiplication. The different operations give different structures – you don’t get the same result if you combine the same two elements using different operations.
Interesting side fact: the use of the symbol ℤ to mean the whole numbers comes from German: it stands for both Zahlen, meaning “numbers”, and zählen, meaning “to count”. This double-letter font is called ‘blackboard bold’, and mathematicians use blackboard bold symbols to represent important concepts – like the ‘counting numbers’, or integers.
There’s a few more things we need to make something definitely a group. For example, in the case of integers under +, we have a special element in the group: 0. This has the property that if you combine it with another element of the group, using the group operation, it doesn’t change the element. So, 7 + 0 = 7. 0 here is called the identity element of the group – it doesn’t change anything. For something to definitely be a group, we need the identity element to exist and be unique in the group.
It’s also more obvious how the structure of (ℤ, +) differs from (ℤ, ×) when you think about identity elements: in (ℤ, +), 0 is the identity, but in (ℤ, ×) the identity element would be the number 1 – if you multiply something by 1, it doesn’t change.
There are a couple more conditions we need to satisfy to be sure we have a group. Since our definition of the integers includes negative numbers, we find that there are some special pairs of elements in the group: elements which we can combine with others and come out with the identity element as the result! 4 + (-4) = 0. We say -4 is the inverse of 4 in the group (ℤ, +), and for something to be a group, you need every element in there to have an inverse (including the identity element – but that should be its own inverse: 0+0=0).
You might have spotted an issue here: (ℤ, +) definitely has this property – whatever number you can name, I can find its inverse easily by putting a minus symbol in front. But this doesn’t work with (ℤ, ×). The inverse of a number N under the operation ×, which we can combine with it to get the identity 1, is given by 1/N. For example, 2 × ½ = 1. But these inverses – which will be fractions, and not whole numbers – are not elements of the group. This means (ℤ, ×) isn’t actually a group! This type of structure, which is less strict than a group, is called a monoid.
We’re not quite done yet: there’s one more property we need our set and operation to have in order to be a group, and that’s called associativity. It’s a slightly funny one to define, but it essentially means that if you combine two elements using the operation, and then combine the result with another element, it doesn’t matter which order you apply the two operations in. That is: (a+b)+c = a+(b+c).
This works for our group (ℤ, +) of integers under addition: (1+2)+3 = 1+(2+3) = 6 whichever way you look at it. With this property of associativity, any set and operation which are closed, have an identity and all elements have inverses, that’s the definition of a group.
Groups in your area
There are plenty of examples of group structures all around us. You probably look at one every day: a clock face. The numbers on a 12-hour clock form a finite set, and you can combine them by adding times: 4 o’clock plus 5 hours is 9 o’clock, and it wraps around: 10 o’clock plus 5 hours is 3 o’clock. You always get something in the group, 12 o’clock is the identity element (12 hours after 5 o’clock, it’s still 5 o’clock), and you can find the inverse of something by counting backwards.
An even simpler example of a group structure can be found on your coffee tray, next to a hot drink: a battenberg cake. In this group, there are two elements: pink cake, and yellow cake, and the definition of the group operation is laid out as a table in the cake’s cross-section. The top row and left column show what happens if you add pink, and the bottom row and right column are what happens if you add yellow. Pink is the identity element: pink + pink = pink, and yellow + pink = pink + yellow = yellow.
The identity is its own inverse, but we also have an inverse for yellow: yellow + yellow = pink (bottom right corner). This has the same structure as one of the simplest groups, and it’s denoted ℤ2. It’s the only group that has two elements – any other group with exactly two elements is equivalent to it.
You might find another example of a group around the house too – a Rubik’s cube, which has an underlying group describing how it moves. The elements of the group are moves on the cube, and they can be combined by simply performing one after the other. This group is also finite, but it’s got more than two elements (it’s actually got 43,252,003,274,480,856,000 elements, but who’s counting) – and because it’s finite, starting from a solved cube and repeating the same move over and over will get you back to a solved state eventually. The fascinating group underlying the Rubik’s cube is the reason why it’s pictured at the top of the Wikipedia page for Group Theory.
What’s Abel got to do with it?
Maybe you’re still wondering what an abelian group is! We’ve seen various properties that groups can have – being finite, or infinite, or having a certain number of elements. Another property possessed by some but not all groups is that of being abelian. To understand this, we need to think about how the group operation works.
A property of some group operations, which might seem so obvious that you may not even notice it, is that it doesn’t matter which way round the two elements are when you combine them. If you want 2 + 4, you can write it as 4 + 2 and you get the same result, and it’s 6 but nobody cares how you got there. And if I take two numbers on a clock and add them the other way round – 5 hours after 3 o’clock, or 3 hours after 5 o’clock, it’s still 8 o’clock whichever way you work it out.
But this doesn’t always work. The moves on a Rubik’s cube give you a different result if you do them in a different order (go and find one now and try it, if you don’t believe me). Multiplying two matrices together – and certain sets of matrices form a group under standard matrix multiplication – gives a different result if you do it in a different order. Some operations give the same result either way round, and we say they commute.
There are some groups in which only some pairs of elements commute – for example, the identity commutes with everything in a group. But if a group has the property that all of its elements commute with all the others, it’s called an abelian group. Group theorist Camille Jordan named them as a nod to Abel’s work on groups of polynomials which commute. Unusually, despite being named after a person, abelian groups aren’t usually spelled with a capital A – showing just how ubiquitous they are.
Abelian groups are generally easier to understand and work with than general groups, and finite abelian groups have been thoroughly studied. Infinite abelian groups, however, are still not completely understood, and are an area of current research. This means mathematicians today are still researching them – and every time a mathematician studies a group with elements which all commute, they remember Niels Henrik Abel.