  The Steinitz class of a number field extension K/k is an ideal class in the
ring of integers O_k of k, which, together with the degree [K:k] of the
extension determines the O_k-module structure of O_K. We call rt(k,G) the
classes which are Steinitz classes of a tamely ramified G-extension of k. We
will say that those classes are realizable for the group G; it is conjectured
that the set of realizable classes is always a group. We define A'-groups
inductively, starting by abelian groups and then considering semidirect
products of A'-groups with abelian groups of relatively prime order and direct
products of two A'-groups. Our main result is that the conjecture about
realizable Steinitz classes for tame extensions is true for A'-groups of odd
order; this covers many cases not previously known. Further we use the same
techniques to determine rt(k,D_n) for any odd integer n. In contrast with many
other papers on the subject, we systematically use class field theory (instead
of Kummer theory and cyclotomic descent).
