Search and witness problems in group theory

Abstract Decision problems are problems of the following nature: given a property and an object , find out whether or not the object has the property . On the other hand, witness problems are: given a property and an object with the property , find a proof of the fact that indeed has the property . On the third hand(?!), search problems are of the following nature: given a property and an object with the property , find something “material” establishing the property ; for example, given two conjugate elements of a group, find a conjugator. In this survey our focus is on various search problems in group theory, including the word search problem, the subgroup membership search problem, the conjugacy search problem, and others.