Counting Schur Rings over Cyclic Groups of Semiprime Order
Channel: Andrew Misseldine
Date: 2020-04-16

Abstract: Schur rings, or S-rings for short, are objects in the field of abstract algebra and are subrings of group rings which fulfill certain conditions. An S-ring is uniquely determined by a partition of elements of the group in which it belongs. Much research has been done to determine which partitions of the group induce a Schur ring structure. With the Fundamental Theorem of Schur rings over Cyclic Groups recently proven by Leung and Man, it is now possible to begin to enumerate all Schur rings over cyclic groups, but this enumeration is deeply affected by the factorization the order of the group. Complete enumeration of Schur rings over the cyclic group of order pn where p is a prime number and n is an positive integer was accomplished by Misseldine. This project will utilize similar combinatorial methods of counting to enumerate all Schur rings over a cyclic group whose order has no repeated prime divisors, e.g. pq where p,q are distinct primes.

Presentation given by Max Sullivan, a mathematics student at Southern Utah University.

This research was funded by the Walter Maxwell Gibson Research Fellowship from the College of Science and Engineering at Southern Utah University.