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

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The stable letter has infinite order

Statement

The stable letter $t$ of an HNN extension has infinite order.

Facts & Assumptions

Given: An HNN extension with stable letter $t$ .

[L1]

Every Britton-reduced HNN word containing a stable letter is nontrivial. (Britton's lemma)

Proof

technique · direct

1.1

For any nonzero integer $n$ , the word $t^{n}$ contains a stable letter and has no base coefficient at which a pin could occur, so it is Britton-reduced.

L1given

2.1

By [L1], the element represented by $t^{n}$ is nonidentity for every $n \neq 0$ . Hence no nonzero power of $t$ equals $1$ , and $t$ has infinite order.

L1step 1.1algebra∎

Depends on

Britton's lemma

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)