Membership problem for a subgroup
= Membership problem for a subgroup
= Generalized word problem
{synonym}
For a specified subgroup $H\le G$ and a chosen alphabet for $G$, the membership problem asks whether an input word represents an element of $H$. Solubility of the ordinary <word problem for a group> does not imply solubility of this problem, even for finitely generated subgroups: a <Mihailova subgroup> of a product of finite-rank <free groups> can have undecidable membership.