group presentation

Think of building a group out of raw material and a list of rules. The raw material is a set of generators — symbols you are free to multiply and invert in every possible way, producing all reduced words. The rules are relations: equations you decree to hold, such as ‘r equals the identity.’ A presentation is exactly this bookkeeping: list the generators, list the relations, and the group is everything those words can build subject to nothing more than the relations and the group axioms force.

Formally, a presentation ⟨ S | R ⟩ denotes the quotient F(S) / N, where F(S) is the free group on the generating set S and N is the normal closure of the relator set R (the smallest normal subgroup containing R). A group G is said to be presented by ⟨ S | R ⟩ when G is isomorphic to this quotient. A presentation is finite when both S and R are finite; a group admitting such is finitely presented, a strictly stronger condition than being finitely generated.

Presentations are flexible but treacherous. The same group has infinitely many presentations, and two presentations can look utterly different yet define isomorphic groups — indeed deciding whether two finite presentations give isomorphic groups (the isomorphism problem) is algorithmically unsolvable in general, as is the word problem of deciding whether a given word equals the identity. So a presentation is a compact description, not a transparent one: it specifies a group without, by itself, revealing the group's properties.