computer science

Todd–Coxeter algorithm

Todd–Coxeter algorithm is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Todd–Coxeter algorithm rather than just read about it. In short: In group theory, the Todd–Coxeter algorithm, created by J. A.

Key takeaways

Todd–Coxeter algorithm belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Todd–Coxeter algorithm to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Todd–Coxeter algorithm from memory before moving on to harder problems.

Reference excerpt

In group theory, the Todd–Coxeter algorithm, created by J. A. Todd and H. S. M. Coxeter in 1936, is an algorithm for solving the coset enumeration problem. Given a presentation of a group G by generators and relations and a subgroup H of G, the algorithm enumerates the cosets of H on G and describes the permutation representation of G on the space of the cosets (given by the left multiplication action). If the order of a group G is relatively small and the subgroup H is known to be uncomplicated (for example, a cyclic group), then the algorithm can be carried out by hand and gives a reasonable description of the group G. Using their algorithm, Coxeter and Todd showed that certain systems of relations between generators of known groups are complete, i.e. constitute systems of defining relations. The Todd–Coxeter algorithm can be applied to infinite groups and is known to terminate in a finite number of steps, provided that the index of H in G is finite. On the other hand, for a general pair consisting of a group presentation and a subgroup, its running time is not bounded by any computable function of the index of the subgroup and the size of the input data.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Todd–Coxeter algorithm

Start with the simplest possible case. Write down what Todd–Coxeter algorithm claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Todd–Coxeter algorithm before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Todd–Coxeter algorithm ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Todd–Coxeter algorithm

In research: Todd–Coxeter algorithm appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Todd–Coxeter algorithm in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Todd–Coxeter algorithm is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational group theory, so understanding it makes those chapters shorter.
In everyday life: Look for Todd–Coxeter algorithm outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Todd–Coxeter algorithm in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Todd–Coxeter algorithm means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Todd–Coxeter algorithm out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Todd–Coxeter algorithm in simple terms?

In group theory, the Todd–Coxeter algorithm, created by J. A.

Why does Todd–Coxeter algorithm matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Todd–Coxeter algorithm?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Todd–Coxeter algorithm.