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Groups today: a whistle-stop tour

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Groups today: a whistle-stop tour

Starting from their origin as an abstraction of the concept of symmetry, groups have become a core part of the language of modern mathematics and theoretical physics. On this page, find out how groups can help describe roots of polynomials, holes on a surface, and even the laws of physics!

What is a group?

In mathematics and physics, an important notion is that of a symmetry: a transformation from an object to itself which leaves all its features unchanged. Across the 19th and early 20th centuries, mathematicians gradually realised that all collections of symmetries had in common a list of key properties:

  1. The ability to compose symmetries to get a new symmetry (closure/associativity)
  2. The ability to undo symmetries (inverses)
  3. The existence of a do-nothing symmetry which leaves the object untouched (identity).

A group abstracts a collection of symmetries, taking these key properties as its starting point. That is, a group is a set of elements with a combining operation that satisfies exactly the three properties listed above.

Find examples of groups and more ways to think about what a group is below:

Groups: Basics — This package introduces what a group is and some basic concepts in group theory.

Nowadays, groups are ubiquitous throughout mathematics and theoretical physics. To describe polynomials, we talk about their Galois group. To describe the topology of shapes, we talk about their fundamental group. And modern theories of physics take the symmetries of the universe as their starting point. Find out more details about how groups are used throughout maths and physics below.

How are groups used in...

Finding the laws of physics

The world around us possesses many symmetries. There is time-translation symmetry: the belief that the laws of physics will be the same tomorrow as they are today; space-translation symmetry: the laws of physics are the same on Mars as they are here on Earth; and so on. 

In the early 20th century, Emmy Noether proved the amazing result that any symmetry of our Universe gives rise to its own law of physics. Collectively, the laws derived from Noether’s theorem are called conservation laws, since each law says that a certain physical quantity is conserved over time. Time-translation symmetry corresponds to conservation of energy, while space-translation symmetry corresponds to the conservation of momentum. Hence understanding the symmetries of our Universe, a job for group theory, allows us to understand the laws of physics. 

Read more about Emmy Noether and her work here:

Emmy Noether and the power of symmetry — Find out about how Emmy Noether forever tied physics together with symmetries.

In fact, most modern formulations of physics take symmetry as their central guiding principle. Rather than formulating a theory first and then looking for its symmetries later, physicists first decide what symmetries their theory should possess and then see how reality fits in with that. The approach has had startling success. Several fundamental particles, including the famous Higgs boson, were predicted to exist based on the assumption that certain (rather abstract) symmetries exist, and only later discovered in experiments. The hope is that symmetry will eventually guide us to the hotly sought after theory of everything. Find out more in the article below:

Symmetry making and symmetry breaking — A closer look at the power of symmetry in modern physics.