Abelian Groups - László Fuchs [Springer]

432 12 Torsion-Free Groups
If we had A D B ˚ C with B; C ¤ 0, alsop-pure in J p , we would obtain the
contradiction A=pA Š B=pB ˚ C=pC Š Z.p/ ˚ Z.p/.
ut
Our next goal is to obtain more explicit examples for indecomposable groups. In
the constructions, we will often use the practice—as a matter of convenience—of
writing p 1 a as an abbreviation for what would properly be written as an infinite
set: fp 1 a;:::;p n a;:::g. Also, we will construct groups A by starting from a basis
fa 1 ;:::;a n g of a Q-vector space V, and then specifying additional generators of A
in V, or by starting from a direct sum, and then adjoining elements from its divisible
hull, even without mentioning V. Unexplained letters, like a n ; b n ; c n , will denote
independent elements in some unspecified Q-vector space; in any case, our notation
should be self-explanatory.
Example 4.2. Every torsion-free group of rank 1 is indecomposable.
Example 4.3 (Bognár [1]). Consider a set fp 1 ::::;p n g of primes, and an integer m relatively prime
to each p i .Foreveryi, letE i Dhpi 1 e i i and G i Dhpi
1 e i ; m 1 e i i be groups of rank 1; clearly,
E i D mG i .DefineA as a subgroup of G 1 ˚˚G n as
A DhE 1 ˚˚E n ; m 1 .e 1 C e 2 /;:::;m 1 .e 1 C e n /i:
To show that A is indecomposable, suppose A D B˚C. Notice that Hom.E i ; G j / D 0 if i ¤ j,since
every element of E i is divisible by every power of p i , while G j does not contain any such elements
¤ 0. Therefore, the groups E i are fully invariant in A, sowehaveE i D .B \ E i / ˚ .C \ E i /.
The group E i (as a rank 1 group) is indecomposable, so either E i B or E i C. Assume e.g.
E 1 B and E i C for some i >1. We can write m 1 .e 1 C e i / D b C c with b 2 B; c 2 C.Then
e 1 D mb; e i D mc, which is impossible, since none of e i is divisible in A by any prime divisor of
m. This means that all of E i are contained either in B or in C, and hence either B D A or C D A.
Example 4.4. Modify the preceding example, choosing a prime q different from the p i ,and
forming the group
G DhE 1 ˚˚E n ; q 1 .e 1 C e 2 /;:::;q 1 .e 1 C e n /i:
The same proof applies to establish the indecomposability of this G.
Rigid Systems In the construction of indecomposable groups, the following
concept is useful. A set fG i g i2I of torsion-free groups ¤ 0 is said to be a rigid
system if
Hom.G i ; G j / Š
(
a subgroup of Q if i D j;
0 if i ¤ j:
That is, groups in a rigid system have no endomorphisms other than multiplications
by rational numbers, and no non-trivial homomorphisms into each other. In
particular, groups in a rigid system have no idempotent endomorphisms ¤ 0; 1,
thus they are necessarily indecomposable. A group G is called rigid if the singleton
fGg is a rigid system.
The simplest example for a rigid system is a set of rank 1 groups.
Proposition 4.5. There exists a rigid system of torsion-free groups of rank 1
consisting of 2 @ 0
groups.