Group with solvable word problem

Definition
A group with solvable word problem is a finitely generated group $$G$$ satisfying the following equivalent conditions:


 * 1) There is a finite generating set and an algorithm that, given any word in terms of the generators, can determine in finite time whether or not that word equals the identity element of $$G$$.
 * 2) For any finite generating set there is an algorithm that, given any word in terms of the generators, can determine in finite time whether or not that word equals the identity element of $$G$$.
 * 3) For any algebraically closed group $$K$$, $$G$$ is isomorphic to some subgroup of $$K$$.

Note that the finite time that the algorithm takes to terminate depends on the word itself. However, since the generating set is finite, there are only finitely many words of a given length, and we can hence obtain a bound on the time the algorithm takes, that depends only on the length of the word. However, that bound may not in general be a computable function of the length.