Abelian Groups - László Fuchs [Springer]

438 12 Torsion-Free Groups
(7) (de Groot) (a) A torsion-free group is purely indecomposable if any two
independent elements have incomparable types.
(b) Let V be a Q-vector space with basis fa 1 ;:::;a n ;:::g. Arrange the linear
combinations b D k 1 a n1 CCk r a nr with n 1 < < n r and integers
0 ¤ k i 2 Z; k 1 2 N; gcdfk 1 ;:::;k r g D 1 in a sequence. To the jth b
in the sequence assign the characteristic .0;:::;0;1;0;:::/ with 1 only
at the jth prime to obtain a subgroup A of V. Prove that A is purely
indecomposable.
5 Pathological Direct Decompositions of Finite Rank Groups
From the point of view of direct decompositions, the existence of large indecomposable
groups is a major difference between torsion and torsion-free groups.
Another striking difference between them is the occurrence of various phenomena
in the direct decompositions of torsion-free groups already in the finite rank case.
The situation is more cumbersome than one might expect, and we shall spend some
time to make it more transparent.
It is evident that a torsion-free group of finite rank decomposes into the direct
sum of a finite number of indecomposable groups. The question we pose is this: is
there any kind of uniqueness in these decompositions? The best way to describe the
answer is to quote Kaplansky who claimed that in this strange part of the subject,
anything that can conceivably happen actually does happen.
We will list several examples of pathological decompositions, involving lots of
computations (in particular, divisibility considerations). Complicated technicalities
are, however, unavoidable in view of the nature of the subject, and unfortunately,
they only provide a very superficial insight into the structure of finite rank torsionfree
groups.
Decompositions with the Same Number of Summands Our first theorem is
a variant of Jónsson’s original counterexample that set the stage for further strange
examples. Recall: for a set P primes, we use P 1 a as an abbreviation of the set
fp 1 a j p 2 Pg.
Theorem 5.1. There are rank 3 torsion-free groups that have many non-isomorphic
direct decompositions into indecomposable summands.
Proof. Let P; Q denote two infinite disjoint sets of primes, and p >3aprimep …
P [ Q.Define
A DhP 1 ai˚hP 1 b; Q 1 c; p 1 .b C c/i
with independent a; b; c.Picks 2 Z with gcdfs; pg D1, s 6 0; ˙1 mod p, andfind
r; t 2 Z satisfying sr pt D 1.Thena 0 D sa C tb; b 0 D pa C rb; c are independent,
and we claim that