Britton's lemma

Fix the HNN extension conventionwhere $C,D\le A$ and $\varphi :C\to D$ is an isomorphism. A word with at least one stable letter is nonidentity if, after combining neighboring base-group factors, it contains neither of the pinches in an HNN extensionIn fact such a reduced word cannot represent any element of the base group $A$. The natural map $A\to G$ is injective.

The normal form theorem for an HNN extension proves this by letting base elements and stable letters permute formal transversal normal strings. A reduced string with stable letters moves the empty string to one with positive stable-letter length, so it cannot act like a base element. This proof also establishes the base injection without assuming it in advance.

Reversing the defining convention to $t^{-1}ct=\varphi (c)$ reverses the two displayed pinch types. Keeping the convention fixed is necessary when applying the lemma to an explicit word.