Infinite splitting in the syzygies of quaternionic groups

Abstract

→ F 0 → R → 0) be a free resolution over the group ring R[Φ] where R is commutative and Φ is finite.The n th syzygy Ω R[Φ] n is the stable class of Im(∂ n ) and has a tree structure with roots which do not extend infinitely downwards.We show that Ω R[Q8p] 3 has infinitely many isomorphically distinct modules at the minimal level when R = Z[C ∞ ] is the integral group ring of the infinite cyclic group and Q 8p is the quaternion group of order 8p where p ≥ 3 is prime.This poses severe difficulties in attempting to solve the D(2) problem of CTC Wall for the groups C ∞ × Q 8p