= Messuage lemma for groups
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In the effective group form used for recognition problems, a <finite group presentation> of $K$ and a word $w$ yield, by an algorithm, a finite presentation of a group $K_w$. If $w=1$ in $K$, then $K_w$ is trivial; if $w\ne1$ in $K$, then the specified generators embed $K$ in $K_w$. One can additionally arrange that the <normal closure> of the image of $w$ is all of $K_w$. The finite presentation can be written without first deciding which case holds. This is the collapse-versus-embedding construction behind the <Adian–Rabin theorem>.
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