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The rank of a free group admitting a finite basis
Definition
A free group has finite rank if it admits a finite free basis. In that case its rank is
where is any finite free basis of . This is well-defined by Any two finite free bases of the same group 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.
Depends on
Used by
- A free group of rank at least two has subgroups of every finite rank Corollary
- The free product of two infinite cyclic groups is the free group on two generators Corollary
- Free abelian groups of rank at least two are not hyperbolic Proposition
- Free groups of rank at least two have exponential growth Theorem
- Non-elementary hyperbolic groups contain a rank-two free subgroup Theorem
- The fundamental group of a finite wedge of circles is free of that rank Theorem
- The Schreier index-rank formula Theorem
Dependency tree · two levels
10 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
- Encyclopedia of Mathematics, Free group (standard reference, not scraped)