Transposition as a Permutation: A Tale of Group Actions and Modular Arithmetic

Abstract

SummaryConverting a matrix from row-order storage to column-order storage involves permuting the entries of the matrix. How can we determine this permutation given only the size of the matrix? Unexpectedly, the solution to this question involves the use of elementary group theory and number theory. This includes the Chinese remainder theorem, finding multiplicative generators modulo pn for prime p, and using these to find orbit generators for a group action, a subgroup of ℤN* acting on all of ℤN.