1 - An elementary introduction to coset table methods in computational group theory

Summary

PROLOGUE

“…; in fact the method can be reduced to a purely mechanical process, which becomes a useful tool with a wide range of application. …, we venture to predict that our method will prove quite practicable for most groups (at any rate such as occur naturally in geometry or analysis) of order less than a thousand, and for many groups of much higher order.”

J.A. Todd, H.S.M. Coxeter, 1936.

The paper ‘A practical method for enumerating cosets of a finite abstract group’ from which the quotation is taken, may very well be thought of as starting the subject of a series of 5 survey lectures which were given at “Groups -St. Andrews 1981” under the title “Computational methods in group theory”. The quotation itself was the guiding principle for them; I neither dealt with the question of algorithmic solubility of problems - this will in fact often be obvious - nor with the use of computers for solving specific group-theoretic problems in an ad hoc fashion but restricted attention to methods which are designed (and have been implemented) for practical use in a variety of cases.

Of course in 1936 Todd and Coxeter proposed and used their method for hand calculations.