Free Groups & Presentations

Reduced words, the universal property, and groups from generators and relations.

The idea

A group presentation specifies a group by naming generators and listing the relations they satisfy, as in $\langle r, s \mid r^{4} = 1,\ s^{2} = 1,\ s r s = r^{-1} \rangle$. To make this precise we first need a group whose generators satisfy no relations at all, so that the relations we list are exactly the relations that hold.

Definition (Free group).

The free group $F(S)$ on a set $S$ has as its elements the reduced words in the symbols of $S$ and their inverses — finite strings with no adjacent $x x^{-1}$ or $x^{-1} x$ — and the product of two words is obtained by writing one after the other and cancelling any such pairs that appear.

On two generators $a$ and $b$ the reduced words fan out in a tree, as the figure shows: each multiplication by a generator steps to a new word, and no path ever returns to an earlier one.

Theorem (Universal property of the free group).

Every map from $S$ into a group $G$ extends to exactly one homomorphism $F(S) \to G$.

Words in $S$ carry no relations the map must respect, which is why the extension always exists. In particular, every group generated by $S$ is a quotient of $F(S)$. The presentation $\langle S \mid R \rangle$ names one such quotient: $F(S)$ modulo the smallest normal subgroup containing the words of $R$, which forces each word of $R$ to equal the identity and forces nothing else.

Ways to work on it

• Walkthrough. Reduced words, the universal property, and reading a presentation.
• Practice. Find the order of a group from a dihedral presentation.
• Hardest. Identify the familiar group that a presentation describes.

Not sure where to start? Take the ten-question placement test.