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For a Dehn presentation, repeatedly freely reduce a word or replace a relator segment longer than half its perimeter by its shorter complementary segment. Length decreases strictly. A word represents the identity exactly when the procedure reaches the empty word: any nonempty terminal null word would contradict the defining Dehn property. The finite list of relators makes each search effective.

A finite group presentation with cyclically reduced symmetrized relators is a Dehn presentation if every nonempty freely reduced null word contains a relator segment longer than half that relator. Replacing the segment by the inverse complementary segment strictly shortens the word. Finiteness of both the alphabet and the relator set is part of this definition; merely permitting all null words as an infinite relator set would make the shortening property vacuous as a finiteness criterion.

A Dehn presentation is a finite group presentation with the following strict shortening property: every nonempty freely reduced word representing the identity contains a consecutive segment $u$ of a relator $r$ satisfyingRelators are taken as cyclically reduced words with a symmetrized relator set: their inverse words and cyclic permutations are included. This convention allows the matched segment to start anywhere on either orientation of a relator. The symmetrized set remains finite and has the same maximum relator length.

If $r=uv$, the relation replaces $u$ by $v^{-1}$, of length $|r|-|u|<|u|$. The Dehn algorithm alternates this replacement with free reduction. Every step shortens the word; it terminates at the empty word exactly for words representing the identity. The strict inequality matters: a half-perimeter match alone need not shorten anything.

Finiteness is part of the convention used here and is needed for the maximum and counting argument in the last part. With an unrestricted infinite relator set, merely imposing the shortening property would not by itself justify that conclusion.

Minimality of the conjugacy representative implies that $w$ is freely reduced; otherwise free reduction produces a shorter representative of the same element. It is also a cyclically reduced word. If $w=sv{s}^{-1}$ as a freely reduced word with its first and last letters inverse, the shorter word $v$ represents a conjugate, contradicting minimality. This is the cyclic reduction of a shortest conjugacy representative.

The word is nonempty because the element has order greater than one. Since it is cyclically reduced, no cancellation occurs between successive copies in $w^n$, and $w^n$ is a nonempty freely reduced word. It represents the identity because conjugation preserves the order. Apply the defining property of the Dehn presentation to obtainThe relator belongs to the finite symmetrized set fixed in the definition. No assumption that the group itself is torsion-free is made.

In a finite Dehn presentation, every nonidentity finite-order element has a shortest conjugacy representative of length at most half the length of some relator. Indeed a positive power contains a Dehn segment. If the representative had length at least $k=\lfloor |r|/2\rfloor +1$, the first $k$ letters of that segment would fit into a cyclic rotation and could be replaced by fewer letters, contradicting minimality. Bounded relator lengths and a finite alphabet therefore give finitely many conjugacy classes of finite-order elements. With no relators the group is free and torsion-free, so only the identity class remains.