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Welcome to the world of symmetry

When mathematicians study symmetry they use the theory of groups. The field underpins much of modern mathematics and is also hugely relevant to other areas — from physics to computer science. And although it's over two hundred years old, group theory is still a vibrant and dynamic area in which exciting things are happening right now.

This is why the Isaac Newton Institute (INI) in Cambridge is running a research programme called Groups, representations and applications (GRA), which focusses on the wealth of new results and directions in the area. To help you share in the fascination of group theory, and understand what the GRA programme is all about, we have put together the following collection of articles and podcasts. Enjoy!

New perspectives in symmetry — We talk to Colva Roney-Dougal and Martin Liebeck, organisers of GRA programme, to find out what it's all about.

The mathematics of shuffling — Believe it or not, but group theory comes in extremely handy when you're shuffling a pack of cards. This article explains why.

The magic of shuffling — Since we're on the topic of shuffling, here's how a little maths goes a long way with card tricks too.

The maths and magic of shuffling: the podcast – In this podcast mathematician Cheryl Praeger and magician Will Houstoun reveal the maths and magic behind shuffling cards.

Maths in three minutes: Groups — Here's a quick introduction from the Plus library to the concept of a group, complete with two easy examples.

The power of groups — What can you do with a group? Colva Roney-Dougal gives a more detailed introduction to groups and an example of their power in this article from the Plus library.

Maths in a minute: Representing groups — Find out how a group can come in many different guises, and why it's useful to find one way to represent them all.

Maths in a minute: Cyclic groups — When things go round and round a cyclic group may be just what you need.

An enormous theorem: The classification of finite simple groups — Enormous is the right word: this theorem's proof spans over 10,000 pages in 500 journal articles! Find out what it says in this article from the Plus library.

Stubborn equations and the study of symmetry — This article from our library tells a tale of impossible equations, two tragic heroes, and the foundation of group theory.

If you'd like more mathematical detail then you can watch talks from the GRA programme on the INI website. To see more Plus articles about groups click here.

We produced this collection of content as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit to find out more.

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