About 20 years ago I was asked a question of Herbert Kociemba, a computer scientist who has one of the best Rubik’s cube solving programs known. Efficient methods of storing permutations in and (the groups of all permutations of the edges and vertices , respectively, of the Rubik’s cube) are needed, hence leading naturally to the concept of the complement of in . Specifically, he asked if has a complement in (this terminology is defined below). The answer is,
as we shall see, ”no.” Nonetheless, it turns out to be possible to introduce a slightly more general notion of a “-tuple of complementary subgroups” (defined below) for which the answer to the analogous question is ”yes.”
This post is a very short summary of a paper I wrote (still unpublished) which can be downloaded here.