# Teacher package: Group theory

Issue 48The *Plus* teacher packages are designed to give teachers (and students) easy access to *Plus* content on a particular subject area. Most *Plus* articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world,
find maths in unusual places, and delve into mathematical history and philosophy. We therefore hope that our teacher packages provide an ideal resource for students working on projects and teachers wanting to offer their students a deeper insight into the world of maths.

## Group theory

This teacher package brings together all *Plus* articles on group theory. In addition to the *Plus* articles, the *try it yourself* section provides links to related problems on our sister site NRICH.

We've grouped the articles into three categories:

- Getting to know groups — The articles in this category serve as thorough introductions to group theory and contain explicit definitions and examples of groups;
- Group applications — Articles in this category explore applications of group theory. The mathematical descriptions here are mostly intuitive, so no previous knowledge is needed.
- Group history — This category focuses on the history of group theory, from its beginnings to recent breakthroughs.

We also present a list of relevant problems from our sister site NRICH.

### Getting to know groups

The power of groups — This article defines groups, gives explicit examples and uses groups to solve a game of Solitaire.

An enormous theorem: The classification of finite simple groups — This article defines groups by examples and explores a problem in group theory that led to the longest proof in mathematical history.

Chaos, chance and money — This article explains how, when it comes to groups, randomness can generate certainty. It contains an explicit example using cyclic groups, but does not require a formal understanding of what a group is.

### Group applications

Ringing the changes — Did you know that church bell ringers have to memorise sequences of several thousand numbers, and that it can take up to 18 hours to translate these sequences into perfect bell ringing? This article explains why, and looks at some group theory behind bell ringing.

Rubik success in 26 steps — The article explains how group theory was used to show that you can solve a Rubik's cube in 26 steps. It contains a description of cosets.

A symmetry approach to viruses — An account of how group theory is used to study the shape of viruses. The article gives no explicit description of groups.

Through the looking glass — Some molecules come in both left and right handed versions, while others are indistinguishable from their reflections. The article explores the role of mathematical symmetry, and therefore group theory, in chemistry. It gives no explicit description of groups.

The search for Higgs — The Higgs boson is the most mysterious of all elementary particles: modern physics does not make sense without it, but no-one has ever seen one. Particles obey certain symmetries and so group theory has turned out to be very important to particle physics. The article gives no explicit description of groups.

### Group history

Genius, stupidity, and genius again — Évariste Galois died at the age of 21 in a duel, but not before laying down the foundations of group theory. This article looks at the story of his short life and at the problems that led him to invent groups. This article introduces the problem of finding the roots of a polynomial, but does not go on to define groups.

Solving symmetry — How do you spot if one group is the same as another? This article reports on a mathematical breakthrough involving one of the most complex groups of all: a group known as *E8*. The article contains an intuitive introduction to groups in general and Lie groups in particular, and touches on some
applications.

Abel for symmetry — This article introduces two mathematicians whose work was instrumental in the understanding of group theory. Contains an intuitive introduction to group theory and the problem of classifying groups, and touches on some applications.

The little lemma that could — the work of Ngô Bào Châu — A look at the work of a 2010 Fields medallist.

A Schock Prize for an enormous theorem — The Rolf Schock Prize in Mathematics 2011 was awarded to Michael Aschbacher for his contribution to the classification of finite simple groups.

Beautiful symmetry provides glimpse into quantum world — A complex symmetric structure known as the exceptional Lie group E8, which has so far only existed in the minds of mathematicians, seems to have turned up in real life for the first time.

### Try it yourself with our sister site NRICH

Here is a selection of relevant problems and articles from our sister site NRICH.

Grouping transformations — An introduction to group theory using transformations of the plane.

Simplifying transformation — This problem involves combining transformations of the plane to see how many different ones can be made. The problems does not explicitly involve groups, but is good for getting a hang of inverses and products.

Shuffles — This interactivity can be used to explore the properties of small groups. The problems Sheffuls and Zodiac use Shuffles to explore permutations.

An introduction to mathematical structure — An elementary introduction to groups with many explicit examples.

Small groups — These puzzles provide an introduction to all groups of order less than five.

Groups of sets — This problem introduces a set of sets and asks you to prove that it forms a group.

What's a group — This problem explores some of the best-known infinite groups.

An introduction to Galois theory — This article provides a basic introduction to Galois theory, exploring the roots of polynomials.