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The free product of an arbitrary family of groups
Definition
For a family , a free product is a group with homomorphisms in the sense of Monoid homomorphism and group homomorphism, such that for every group and every family of homomorphisms , there is a unique homomorphism satisfying for all . It is denoted . Injectivity of the maps is not part of this definition.
Depends on
Used by
- Amalgamation over the trivial group is the ordinary free product Corollary
- Each factor is a retract of a free product when all other factors are sent trivially Corollary
- Free products are unique up to a unique factor-compatible isomorphism Corollary
- If one set in a van Kampen cover is simply connected, the other fundamental group surjects with overlap-generated kernel Corollary
- An HNN extension with its stable letter Definition
- The canonical surjection from a free product to the direct product of its factors Example
- Conventions and proved scope for free products and amalgamation Remark
- A free product has the union presentation of presentations of its factors Theorem
- A group pushout is the quotient of a free product by the amalgamating relations Theorem
- Free groups on disjoint bases freely multiply to the free group on their union Theorem
- Kurosh subgroup theorem Theorem
- Normal form theorem for free products Theorem
- Normal forms in an HNN extension are unique relative to chosen transversals Theorem
- Reduced syllable words form the free product of a family of groups Theorem
Dependency tree · two levels
5 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
- George D. Torres, Combinatorial Group Theory, §2 (standard reference, not scraped)
- B. H. Neumann, Lectures on Topics in the Theory of Infinite Groups, Ch. 9 (standard reference, not scraped)