# Free Group

**Type:** Concept  
**Domain:** algebra  
**Codex URL:** /mathematics/algebra/free-group/  
**Entry status:** Live — v1.0 (2026-07-09)

## Summary
The most unconstrained group on given generators — no relations beyond group axioms. Universal property. Nielsen-Schreier: subgroups of free groups are free. Word problem undecidable for general presentations. Banach-Tarski connection via F₂ ⊂ SO(3).

## Sources
- [Tier 1] Lyndon, R.C. and Schupp, P.E. (2001). Combinatorial Group Theory. Springer.
- [Tier 1] Serre, J.-P. (1980). Trees. Springer.
- [Tier 2] Magnus, W., Karrass, A. and Solitar, D. (2004). Combinatorial Group Theory. Dover.
- [Tier 3] Stillwell, J. (1993). Classical Topology and Combinatorial Group Theory. 2nd ed.

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*Mathematics Codex entry v1.0 — added 2026-07-09 — thecodex.expert/mathematics/*