Abelian Groups - László Fuchs [Springer]

444 12 Torsion-Free Groups
A D B i ˚ C i with C i DhX i ˚ Y; p 1 .k i x i C y/i (12.11)
where the C i ’s are non-isomorphic (and indecomposable). Then to ensure that k i x i C
y D k i r i b 1 C .k i t i 1/x 1 C .x 1 C y/ will be divisible by p, we must select k i such
that
k i r i 0 and k i t i 1 mod p: (12.12)
For any k i with 12m 1, then we guarantee that
k j 6 ˙k i mod p for i ¤ j, and therefore, no two of the groups C 1 ;:::;C m are
isomorphic.
ut
This theorem also shows that, for any rank r 3, and for every integer m, there
exist torsion-free groups of rank r which have at least m pairwise non-isomorphic
decompositions into indecomposable summands. However, Lady’s theorem 6.9 will
show that no torsion-free group of finite rank admits infinitely many non-isomorphic
decompositions.
Possible Rank Distributions The question of rank distributions in direct
sums of finite rank torsion-free groups is too important to leave out completely.
The problem was completely solved by Yakovlev and Blagoveshchenskaya in a
satisfactory way. The proof is too long, it requires further structural information
to be determined, so here we cannot do more than just stating the theorem without
proof. As expected, the rank 1 summands play a special role.
Theorem 5.6 (Yakovlev [1], Blagoveshchenskaya–Yakovlev [1]). Let
n D r 1 CCr k D s 1 CCs`
be two partitions of the positive integer n 3.r i ; s j 2 N/. Let u and v denote the
number of r i and s j , respectively, that are equal to 1.
A necessary and sufficient condition for the existence of a torsion-free group of
rank n to admit a direct decomposition into indecomposable summands of ranks r i ,
and also one with indecomposable summands of ranks s j , is that
(i) r i n v for all i, and s j n u for all j; and
(ii) if r i D n v for some i, then exactly one s j ¤ 1.and is equal to n v/, and if
s j D n u for some j, then exactly one r i ¤ 1.and is equal to n u/. ut
The proof of sufficiency is based on a clever construction of special indecomposable
groups, called flower groups.