Composition factors of the functor of the complex characters

Abstract

When we consider CRC as a map sending any finite group G to the complex vector space CRC(G) of complex valued class functions on G, it becomes an A-fibered biset functor for any group A ≤ C×. Its structure is known for tirivial fiber groups A = 1 and A = C×. While it is a direct sum of simple biset functors in the case that A = 1, in the other case it is a simple C×-fibered biset functor. We noticed that as the fiber group grows, some of simple summands of 1-fibered biset functor CRC unite and form new fibered simple summands. In this thesis, we investigate the structure of the functor CRC for two intermediate fiber groups. The first one is a group containing all pn-th roots of unity for any n ∈ N and for any prime number p from a fixed set of primes π. The second one is the group of all pn-th roots of unity for a fixed n ∈ N. For both cases, we identify its new fibered simple summands by determining uniting summands via defining equivalence relations on them.