Word problem for groups

In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group ${\displaystyle G}$ is the algorithmic problem of deciding whether two words in the generators represent the same element of ${\displaystyle G}$. The word problem is a well-known example of an undecidable problem.

If ${\displaystyle A}$ is a finite set of generators for ${\displaystyle G}$, then the word problem is the membership problem for the formal language of all words in ${\displaystyle A}$ and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on ${\displaystyle A}$ to the group ${\displaystyle G}$. If ${\displaystyle B}$ is another finite generating set for ${\displaystyle G}$, then the word problem over the generating set ${\displaystyle B}$ is equivalent to the word problem over the generating set ${\displaystyle A}$. Thus one can speak unambiguously of the decidability of the word problem for the finitely generated group ${\displaystyle G}$.

The related but different uniform word problem for a class ${\displaystyle K}$ of recursively presented groups is the algorithmic problem of deciding, given as input a presentation ${\displaystyle P}$ for a group ${\displaystyle G}$ in the class ${\displaystyle K}$ and two words in the generators of ${\displaystyle G}$, whether the words represent the same element of ${\displaystyle G}$. Some authors require the class ${\displaystyle K}$ to be definable by a recursively enumerable set of presentations.