Double cosets and searching small groups

Abstract

Double cosets are an important concept of group theory. Although the desirability of algorithms to compute double cosets has been recognized, there has not appeared any algorithm in the literature. The algorithm which we present is a variant of Dimino's algorithm for computing a list of elements of a small group. (By “small” we mean groups of order less than 104, whose list of elements we can explicitly store.) The paper focusses on the problem of searching a small group for elements with a given property. For the record we present Dimino's algorithm and a general algorithm for searching a small group. These two algorithms are not original. We analyse the search algorithm and discuss the role of double cosets in searching. The use of double cosets in the search algorithm does not appear to lead to an improvement over the use of right cosets.