Computational Group Theory

Computational group theory has a history going back more than 80 years. It is a collaborative e ort of researchers in a wide range of areas in both mathematics and computer science. The eld has produced important algorithmic ideas and large software packages that have been used to obtain valuable results in mathematics and in other disciplines. The algorithmic achievements include the development of complicated algorithms designed to have the best possible asymptotic complexity and algorithms of exponential or perhaps unknown complexity that are nevertheless extremely useful in actual computations. There are also techniques for attempting to study instances of problems that are known not have have algorithmic solutions in general. This article is intended as a survey of the eld for the nonexpert. Thus it is written to provide a mathematician, physicist, or chemist who has encountered a particular group with guidance about the kinds of information one should expect to be able to determine about the group and in some cases to o er insight into the size of problems that can be attacked with current hardware. The article also is designed to show the computer scientist unfamiliar with the eld the rich variety of problems concerning the complexity of group-theoretic algorithms. The reader is introduced to several data structures that are unique to computational group theory. These include bases and strong generating sets for permutation groups and coset tables used to describe nitely generated subgroups of free groups. Other themes are the role of the classi cation of nite simple groups in the development of the subject and e orts to cope with the fact that many interesting problems do not have general algorithmic solutions. A number of other surveys of computational group theory have appeared