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.
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The factor maps into a free product with amalgamation are injective
Statement
The canonical maps and are injective.
Facts & Assumptions
Given: The objects and hypotheses in the statement.
Every element of has a unique normal form relative to fixed transversals. A normal word of positive length is nonidentity. The represented group and these conclusions are independent of the chosen transversals. (Normal form theorem for free products with amalgamation).
A group homomorphism is injective if and only if its kernel is trivial. For a group homomorphism , is injective exactly when . (A group homomorphism is injective if and only if its kernel is trivial).
Proof
A nonidentity factor element rewrites either as a nontrivial length-zero element of or as a normal word with one nonidentity transversal syllable.
Neither form is the identity by the normal-form theorem, so each canonical map has trivial kernel and is injective. This includes trivial factors and the case where is a whole factor.
Depends on
Used by
- The two factor images intersect exactly in the amalgamated subgroup Corollary
- Amalgamating C₂ inside C₄ and C₆ gives the presentation with a²=b³ Example
- Amalgamating infinite cyclic groups by multiplication by m and n gives the presentation with relation xᵐ=yⁿ Example
- FALSE: Kurosh says every subgroup of a free product is free False statement
- Free products and suitable amalgamated free products have solvable word problem 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
- 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)