= Boone-Higman theorem
{c}
= Algebraic characterization of soluble word problem
{synonym}
A finitely generated <group> $G$ has a soluble <word problem for a group> if and only if there are a <simple group> $S$ and a <finitely presented group> $P$ with embeddings $G\hookrightarrow S\hookrightarrow P$. The theorem does not require $S$ itself to be finitely presented. In particular, a finitely generated subgroup of a finitely presented simple group has soluble word problem. The primary statement is Theorem I of https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/an-algebraic-characterization-of-groups-with-soluble-word-problem1/748FFD937CEA3727E812C8CDB9CC59B9[An algebraic characterization of groups with soluble word problem].
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