= Britton's lemma
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= Britton lemma
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Fix the <HNN extension> convention
$$
G=\langle A,t\mid tct^{-1}=\phi(c),\ c\in C\rangle,
$$
where $C,D\leq A$ and $\phi:C\to D$ 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 extension>
$$
tct^{-1}\quad(c\in C),\qquad t^{-1}dt\quad(d\in D).
$$
In 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 $t^{-1}ct=\phi(c)$ reverses the two displayed pinch types. Keeping the convention fixed is necessary when applying the lemma to an explicit word.
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