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Amalgamating infinite cyclic groups by multiplication by m and n gives the presentation with relation x^m=y^n

Example

For positive integers $m, n$ , amalgamating infinite cyclic groups $⟨ x ⟩$ and $⟨ y ⟩$ along maps sending a generator to $x^{m}$ and $y^{n}$ gives $⟨ x, y ∣ x^{m} = y^{n} ⟩ .$ Both cyclic factor maps remain injective.

Facts & Assumptions

Given: The objects and hypotheses in the example.

[L1]

Let $G = ⟨ X ∣ R ⟩$ and $H = ⟨ Y ∣ S ⟩$ with disjoint generators, and let $f, h$ embed $K$ . If $T$ generates $K$ and words $u_{t} (X), v_{t} (Y)$ represent $f (t), h (t)$ , then $G *_{K} H ≅ ⟨ X ⊔ Y ∣ R \cup S \cup {u_{t} v_{t}^{- 1} : t \in T} ⟩ .$ (A free product with amalgamation has the factor presentations plus the amalgamating relations).

[L2]

The canonical maps $G \to G *_{K} H$ and $H \to G *_{K} H$ are injective. (The factor maps into a free product with amalgamation are injective).

[L3]

Every class in $W (X) / \sim$ contains exactly one reduced word. (Every class in $W (X) / \sim$ contains exactly one reduced word).

[L4]

For every set $X$ , the group $F_{w o r d} (X) = W (X) / \sim$ together with $i_{w o r d} (x) = [x]$ is a free group on $X$ in the sense of def-free-group. (The word-quotient group $W (X) / \sim$ satisfies the universal property of the free group on $X$ ).

[L5]

Let $F (X)$ be a free group and let $R \subseteq F (X)$ be a set of words, called relations. The group with presentation $⟨ X ∣ R ⟩ : = F (X) / ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)}$ is the quotient by the normal closure of $R$ . The members of $X$ are its generators. In this quotient, every relation in $R$ becomes the identity, as do all consequences forced by normality. (Group presentation by generators and relations).

[L6]

Let $G$ be a group and $R \subseteq G$ . Then $⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{G} = {g_{1} r_{1}^{ε_{1}} g_{1}^{- 1} \dots g_{n} r_{n}^{ε_{n}} g_{n}^{- 1} : n \in N, g_{i} \in G, r_{i} \in R, ε_{i} \in {1, - 1}} .$ For $n = 0$ the displayed product is the identity. Replacing every conjugator $g_{i}$ by $g_{i}^{- 1}$ gives the equivalent convention $g_{i}^{- 1} r_{i}^{ε_{i}} g_{i}$ . (The normal closure of $R$ is the set of finite products of conjugates of elements of $R$ and their inverses).

Verification

technique · direct

1.1

The one-generator empty-relator word model is infinite cyclic: its reduced words are the distinct powers of its generator, and the word-quotient freeness theorem gives the singleton universal property.

givenL1L2L3L4L5L6

2.1

The maps from the edge group are injective because $m, n > 0$ and the factors have infinite order.

step 1.1

3.1

The amalgamated-presentation theorem adds exactly the relation $x^{m} y^{- n} = e$ , equivalently $x^{m} = y^{n}$ , and the factor-embedding theorem preserves both cyclic factors.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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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)