{{Short description|Problem in finite group theory}}
In [[mathematics]], especially in the area of [[abstract algebra]] known as [[combinatorial group theory]], the '''word problem''' for a [[finitely generated group]] ''<math>G''</math> is the algorithmic problem of deciding whether two words in the generators represent the same element. More precisely, if ''<math>A''</math> is a finite set of [[Generating set of a group|generators]] for ''<math>G''</math> then the word problem is the membership problem for the [[formal language]] of all words in ''<math>A''</math> and a formal set of inverses that map to the identity under the natural map from the [[free monoid with involution]] on ''<math>A''</math> to the group ''<math>G''</math>. If ''<math>B''</math> is another finite generating set for ''<math>G''</math>, then the word problem over the generating set ''<math>B''</math> is equivalent to the word problem over the generating set ''<math>A''</math>. Thus one can speak unambiguously of the decidability of the word problem for the finitely generated group ''<math>G''</math>.
The related but different '''uniform word problem''' for a class ''<math>K''</math> of recursively presented groups is the algorithmic problem of deciding, given as input a [[presentation of a group|presentation]] ''<math>P''</math> for a group ''<math>G''</math> in the class ''<math>K''</math> and two words in the generators of ''<math>G''</math>, whether the words represent the same element of ''<math>G''</math>. Some authors require the class ''<math>K''</math> to be definable by a [[recursively enumerable]] set of presentations.
