Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
A free basis of a group
Definition
Let be a group and let . Write for the inclusion. The subset is a free basis of if is a free group on the set in the sense of Free group on a set of generators. Equivalently, for every group and every function , there is a unique group homomorphism whose restriction to is .
Depends on
Used by
- A free group of rank at least two has subgroups of every finite rank Corollary
- With respect to a free basis, the word length of an element is the length of its reduced word Corollary
- The Cayley graphs of ℤ/2 for {1} and of ℤ for {-1,1} are trees, and neither generating set is free Counterexample
- The rank of a free group admitting a finite basis Definition
- The Cayley graph of the free group on two generators is the tree in which every vertex has four neighbours Example
- Any two finite free bases of the same group have the same cardinality Theorem
- Every finitely generated subgroup of a finite-rank free group is a free factor of a finite-index subgroup Theorem
- If no product of two members of a generating set is the identity and the Cayley graph is a tree, the set is a free basis Theorem
- The Cayley graph of a free group with respect to a free basis is a tree Theorem
- Under the stated choice boundary, every subgroup of a free group is free with its nontrivial Schreier generators as a basis Theorem
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, §1.3 (standard reference, not scraped)
- Encyclopedia of Mathematics, Free group (standard reference, not scraped)