We now study how to describe groups in terms of generators and relations, and how to visualize their structure via Cayley graphs. This connects algebra to combinatorics and geometry, and leads naturally to the braid group and .
12.1.1 The Free Group
Recall from Lecture 3: the free group on generators consists of all reduced words (where ,,, and ), with multiplication by concatenation followed by reduction (canceling adjacent inverse pairs and combining adjacent powers of the same generator).
is the “largest” group with generators: every other group generated by elements is a quotient of . If is generated by , define a surjective homomorphism by .
Definition 12.1.1 (Finitely presented group). A group is finitely presented if the kernel of is the smallest normal subgroup of containing some finite set , i.e., (the normal closure: the subgroup generated by all conjugates ). We write where the are generators and the are relations.
Example 12.1.1 (Presentations of familiar groups).
(all generators commute).
where ,.
where .
Note. Given generators , finding relations such that these are a complete set (i.e., they generate the full kernel) is nontrivial. If the relations hold in , then induces a surjection . This is an isomorphism iff the relations generate the entire kernel of .
12.1.2 The Word Problem
Definition 12.1.2 (Word problem). Given a presentation , the word problem asks: given a word in the generators, does represent the identity in ?
Note. For groups where we can “calculate” (e.g., matrix groups, permutation groups), the word problem is trivially solvable: just evaluate the word. But for abstractly presented groups, this can be undecidable. Novikov (1955) and Boone (1959) independently showed there exist finitely presented groups with unsolvable word problem. Two useful tools for groups where the word problem is solvable: Cayley graphs and normal forms.