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CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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 FF has finite rank if it admits a finite free basis. In that case its rank is rank⁡(F):=∣B∣,\operatorname{rank}(F):=|B|, where BB is any finite free basis of FF. 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 xx and yy; their tagged singleton bases have union {x,y}\{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

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 47 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources