this says that since the cube of a is the identity element of G, so is the cube of the inverse of a. Under these conditions the word problem becomes easy. First reduce strings to the empty string, a, aa, A or AA. Then note that we may also multiply by aaa, so we can convert A to aa and convert AA to a. The result is that the word problem, here for the cyclic group of order three, is solvable. This is not, however, the typical case. For the example, we have a canonical form available that reduces any string to one of length at most three, by decreasing the length monotonically. In general, it is not true that one can get a canonical form for the elements, by stepwise cancellation. One may have to use relations to expand a string many-fold, in order eventually to find a cancellation that brings the length right down. The upshot is, in the worst case, that the relation between strings that says they are equal in G is an Undecidable problem. Examples The following groups have a solvable word problem: Automatic groups, including: Finite groups Negatively curved (aka. hyperbolic) groups Euclidean groups Coxeter groups Braid groups Geometrically finite groups Finitely generated free groups Finitely generated free abelian groups Polycyclic groups Finitely generated recursively absolutely presented groups, including: Finitely presented simple groups. Finitely presented residually finite groups One relator groups (this is a theorem of Magnus), including: Fundamental groups of closed orientable two-dimensional manifolds. Combable groups Autostackable groups Examples with unsolvable word problems are also known: Given a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨a,b,c,d | anban = cndcn : n ∈ A⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble Every finitely generated group with a recursively enumerable presentation and insoluble word problem is a subgroup of a finitely presented group with insoluble word problem The number of relators in a finitely presented group with insoluble word problem may be as low as 14 or even 12. An explicit example of a reasonable short presentation with insoluble word problem is given in Collins 1986: Partial solution of the word problem The word problem for a recursively presented group can be partially solved in the following sense: Given a recursive presentation P = ⟨X|R⟩ for a group G, define: then