Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-11
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The free product of two infinite cyclic groups is the free group on two generators

Statement

The free product of two infinite cyclic groups is the free group on two generators, hence has rank two.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

A free product of a family of infinite cyclic groups is a free group on one chosen generator from each factor. The empty family gives the free group on the empty set. (A free product of copies of the infinite cyclic group is a free group).

[L2]

A free group F has finite rank if it admits a finite free basis. In that case its rank is rank⁡(F):=∣B∣, where B is any finite free basis of F. This is well-defined by thm-finite-free-bases-have-the-same-cardinality. This definition is deliberately restricted to free groups that admit a finite free basis. It neither defines rank for a free group whose bases are infinite nor asserts that arbitrary infinite free bases have the same cardinality. (The rank of a free group admitting a finite basis).

Proof

technique · direct
1.1

Apply the preceding result to two factors with chosen generators x and y; their tagged singleton bases have union {x,y}.

givenL1L2
2.1

The resulting free group is free on this two-element set, which is exactly rank two by definition.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

9 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