nLab free group

Idea

The free group on a given set $S$ is the free object on $S$ in the category of groups. The elements of $S$ are called the generators of this group.

Definition

In type theory

In dependent type theory, the free group on a type $A$ is the 0-truncation of the underlying type of free infinity-group of $A$, $\mathrm{FreeGroup}(A) \coloneqq [\mathrm{UTFIG}(A)]_0$. This is because in any dependent type theory with uniqueness of identity proofs, where every type is 0-truncated, the underlying type of the free infinity-group of $A$ is the free group of $A$.

In addition, one could construct the free group directly from the traditional axioms of a group, as detailed below:

The inference rules for free group types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{FreeGroup}(A) \; \mathrm{type}}$$

Introduction rules for free groups:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \eta:A \to \mathrm{FreeGroup}(A) \; \mathrm{type}} \quad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mu:\mathrm{FreeGroup}(A) \times \mathrm{FreeGroup}(A) \to \mathrm{FreeGroup}(A)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \epsilon:\mathrm{FreeGroup}(A)} \quad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \iota:\mathrm{FreeGroup}(A) \to \mathrm{FreeGroup}(A)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A) \quad \Gamma \vdash y:\mathrm{FreeGroup}(A) \quad \Gamma \vdash z:\mathrm{FreeGroup}(A)}{\Gamma \vdash \alpha(x, y, z):\mathrm{Id}_{\mathrm{FreeGroup}(A)}(\mu(x, \mu(y, z)),\mu(\mu(x, y), z))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A)}{\Gamma \vdash \lambda_\epsilon(x):\mathrm{Id}_{\mathrm{FreeGroup}(A)}(\mu(\epsilon, x), x)} \quad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A)}{\Gamma \vdash \rho_\epsilon(x):\mathrm{Id}_{\mathrm{FreeGroup}(A)}(\mu(x, \epsilon), x)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A)}{\Gamma \vdash \lambda_\iota(x):\mathrm{Id}_{\mathrm{FreeGroup}(A)}\mu(\iota(x), x), \epsilon)} \quad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A)}{\Gamma\vdash \rho_\iota(x):\mathrm{Id}_{\mathrm{FreeGroup}(A)}(\mu(x, \iota(x)), \epsilon)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \tau:\mathrm{isSet}(\mathrm{FreeGroup}(A))}$$

Properties

• Every subgroup of a free group is itself a free group. This is the Nielsen-Schreier theorem.

Related concepts

• free monoid

• free product of groups

• free abelian group, free groupoid

• underlying type of free infinity-group

• Fox derivative

References

Early discussion of (subgroups of) free groups (proof of the Nielsen-Schreier theorem) and generalization to the amalgamated free product of groups:

• Otto Schreier, Die Untergruppen der freien Gruppen, Abh. Math. Semin. Univ. Hambg. 5 (1927) 161–183 [doi:10.1007/BF02952517]

Expository review:

• Abhay Chandel, Free groups and amalgamated product, BSc thesis (2013) [pdf, pdf]

Discussion of free groups in homotopy type theory:

• Univalent Foundations Project, §6.11 of: Homotopy Type Theory – Univalent Foundations of Mathematics, Institute for Advanced Study (2013) [web, pdf]

• Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: §6.2 in: Symmetry (2021) [pdf]

• Nicolai Kraus, Thorsten Altenkirch, Free Higher Groups in Homotopy Type Theory, LICS ‘18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, (2018) [arXiv;1805.02069, doi:10.1145/3209108.3209183]