CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Finite-index subgroups of finitely presented groups are finitely presented

Statement

Every finite-index subgroup of a finitely presented group is finitely presented.

Facts & Assumptions

Given: A finitely presented group $G$ and a finite-index subgroup $H \leq G$ .

[L1]

A finite presentation has finite generating and relator sets (Relators and relations; finitely generated, finitely related, and finite presentations).

[L2]

Reidemeister-Schreier presents a subgroup $H$ by finitely many rewritten Schreier generators and relators $τ (t r t^{- 1})$ (The Reidemeister-Schreier presentation theorem).

Proof

technique · direct

1.1

Choose a finite presentation $G = ⟨ X ∣ R ⟩$ and a finite right transversal $T$ for $H$ . By [L1], the sets $X$ and $R$ are finite, so the sets of pairs $(t, x)$ with $t \in T$ , $x \in X ⊔ X^{- 1}$ and $(t, r)$ with $t \in T$ , $r \in R$ are finite.

L1givenchoose

2.1

By [L2], the subgroup $H$ has a presentation whose generators are among the Schreier generators $s (t, x)$ and whose relators are the rewritten words $τ (t r t^{- 1})$ . Step 1.1 shows that both families are finite. Therefore $H$ is finitely presented.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Roger C. Lyndon and Paul E. Schupp, Combinatorial Group Theory (standard reference, not scraped)