Wreath Product Finite p-Groups and their Subgroups

Participants: J. M. Riedl (Theo and Applied Math) and REU students.

Much of my current research is related to the study of the subgroups of iterated wreath product finite p-groups and their automorphism groups. In particular, I am working on classification problems involving the monolithic subgroups H of groups of the form P=(A wr B) or P=((A wr B) wr C), where A, B, and C are cyclic groups of prime-power order, and X wr Y denotes the regular wreath product of X by Y. I also study the automorphism groups Aut(H) of these monolithic subgroups H, and their relationships with the automorphism group Aut(P). Monolithic groups, defined as those having a unique minimal normal subgroup, are of interest from the point of view of representation theory.

I have developed a detailed approach to these classification problems (based on representation theory), and I have successfully implemented this approach in some important cases. This approach includes the task of obtaining a full description and list of all the submodules of the regular module for the group ring RG, where R denotes the ring of integers modulo a power of a prime p, and G denotes an abelian p-group, usually either cyclic or elementary abelian. I call this the Group Ring Submodule Problem. Even for small primes p, this is an extensive computational problem.

During the summers of 2005 and 2006, I directed a research project for several undergraduate students in an NSF-funded Research Experience for Undergraduates (REU) program at the University of Akron. Using techniques that I had previously developed, several of these students (Jennifer Feder of Washington University, Sarah Bockting of the University of Evansville, Arran Hamm of Wake Forest University, and Sam Ruth of Northwestern University) produced a virtually complete solution of the Group Ring Submodule Problem in the case that R is the ring of integers modulo 5 and G is elementary abelian of order 25. The total was just under 600,000 submodules.

Currently I am supervising Lynn Adams, a Master's student at the University of Akron, as she works on the Group Ring Submodule Problem in the case that R is the ring of integers modulo 3, and G is elementary abelian of order 27.

Funding:

• $162,921, Research Experience for Undergraduates (REU), National Science Foundation, Spring 2004. Eight-week summer program at UA in 2005, 2006, 2007.