This is a continuation of our study [A stably free nonfree module and its relevance for homotopyclassification, case ,
Algebr Geom Topol 5 (2005) 899–910] of a family of projective modules over
,
the generalized quaternion (binary dihedral) group of order
. Our approach is
constructive. Whenever
is odd, this work provides examples of stably free nonfree modules of rank
,
which are then used to construct exotic algebraic
–complexes
relevant to Wall’s D(2)–problem. While there are examples of stably free nonfree modules for many
infinite groups ,
there are few actual examples for finite groups. This paper offers an
infinite collection of finite groups with stably free nonfree modules
, given as
ideals in the group ring. We present a method for constructing explicit stabilizing isomorphisms
described
by
matrices. This makes the subject accessible to both theoretical and computational
investigations, in particular, of Wall’s D(2)–problem.
Keywords
exotic algebraic 2-complex, Wall's D(2)-problem, stably
free nonfree module, stabilizing isomorphism, homotopy
classification of 2-complexes, truncated free resolution,
generalized quaternion groups, single generation of
modules, units in factor rings of integral group rings
