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
Currently I am supervising Lynn Adams, a Master's student
$162,921, Research Experience for Undergraduates (REU), National Science Foundation, Spring 2004. Eight-week summer program at UA in 2005, 2006, 2007.