Group
Concept · Mathematics, stage 1: Learn the vocabulary
A group is a set together with one operation that combines any two members to give a third, subject to three conditions: the operation is associative, there is an identity element that changes nothing, and every element has an inverse that undoes it. The symmetries of any object form a group, since rotations and reflections can be combined, done in any grouping, and undone. Because the definition never mentions numbers, a theorem proved about groups applies at once to every structure that satisfies it.
Why it matters
It is the first and clearest case of mathematics studying a pattern of behaviour rather than a kind of object, which is how most of modern algebra works.
Part of
Sources
- Group (mathematics) Wikipedia
- Abstract groups MacTutor History of MathematicsHow the group concept was extracted from work on equations and geometry.
