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In [[mathematics]], especially in the area of [[abstract algebra]] known as [[combinatorial group theory]], the '''word problem''' for a [[finitely generated group]] ''G'' is the algorithmic problem of deciding whether two words in the generators represent the same element. More precisely, if ''A'' is a finite set of [[Generating set of a group|generators]] for ''G'' then the word problem is the membership problem for the [[formal language]] of all words in ''A'' and a formal set of inverses that map to the identity under the natural map from the [[free monoid with involution]] on ''A'' to the group ''G''. If ''B'' is another finite generating set for ''G'', then the word problem over the generating set ''B'' is equivalent to the word problem over the generating set ''A''. Thus one can speak unambiguously of the decidability of the word problem for the finitely generated group ''G''.
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