Search and witness problems in group theory

TL;DR

This paper surveys various search problems in group theory, focusing on how to find proofs or witnesses for properties of group elements and subgroups, highlighting their significance in understanding computational aspects of group theory.

Contribution

It provides a comprehensive overview of different search problems in group theory, clarifying their roles and relationships with decision and witness problems.

Findings

Classifies key search problems like word search and conjugacy search

Explains the importance of these problems in computational group theory

Highlights open questions and research directions

Abstract

Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P. On the other hand, witness problems are: given a property P and an object O with the property P, find a proof of the fact that O indeed has the property P. On the third hand(?!), search problems are of the following nature: given a property P and an object O with the property P, find something "material" establishing the property P; 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.