What is Abstract Algebra?

By Edric Wang, Australian National University (University of Sydney)

When we learn algebra in school we learn about sets of numbers, such as the integers: 0, 1, -1, 2, -2 and so on. We also learn to apply operations on those numbers, such as addition. We learn rules for these operations and consequences of these rules. In abstract algebra, we consider the same ideas in a more abstract setting: instead of studying only numbers, we study how the properties of how abstract elements interact under operations.

For example, let our elements be the symmetries of a square: that is, all of the motions of the square which preserve its shape. For example, reflection about a given axis and clockwise rotation by 90 degrees are two examples of symmetries. Now we can define an operation on these symmetries known as composition: we compose two symmetries by performing one after the other. Now we have a set of elements and an operation on those elements, so we can study the properties of this operation. In this case, these properties will tell us about the symmetry of the square.

It turns out that the two systems we have described (the integers with addition and the symmetries of the square with composition) are both examples of what is known as a group in abstract algebra. This is to say that they share the same underlying structure. Therefore, if we study such groups in the abstract, then any results we obtain can be applied to both the integers and the symmetries of the square. So, abstract algebra allows us to prove results in sweeping generality instead of having to prove essentially the same concept under different guises. In this way, once we recognise a mathematical object as an instance of a certain algebraic structure, we already know a lot about that object without having done any work.

 

Edric Wang was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.

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