Abelian Groups - László Fuchs [Springer]

454 12 Torsion-Free Groups
(3) (Warfield) Suppose A has the 2-substitution property. If A ˚ B Š A ˚ C,andif
B has a summand Š A,thenB Š C.
(4) (Warfield) If A has the n-substitution property, then so does its endomorphism
ring E as a left E-module.
8 Finite Rank p-Local Groups
By a p-local group is meant a group that is also a Z .p/ -module; equivalently, a group
that is uniquely q-divisible for every prime q ¤ p. Thus there is only one prime to
deal with in p-local groups. The restriction to p-local groups has quite a simplifying
effect on the group structure; for instance, there are only two isomorphy classes of
rank 1 torsion-free p-local groups: Z .p/ and Q. Furthermore, like p-groups, p-local
groups may contain non-zero q-basic subgroups only for q D p, so we may call it
simply a basic subgroup.
Preliminaries on Local Groups We begin with important informations on
p-local groups.
Lemma 8.1. Let A be a p-local torsion-free group of finite rank. Then p End A
is contained in the Jacobson radical J of the endomorphism ring End A. Thus
.End A/=J is a finite semi-simple ring of characteristic p.
Proof. We show that for every 2 End A, 1 p is an automorphism of A. It
is clear that if p does not divide a 2 A, then it does not divide .1 p/a either.
Hence multiplication by 1 p preserves p-heights. Therefore, the pure subgroup
Ker.1 p/ must be 0, thus 1 p is monic, and Im.1 p/ is an essential subgroup
of A. From the preservation of heights it follows that Im.1 p/ is pure in A, whence
Im.1 p/ D A. Consequently, 1 p is an automorphism, indeed. ut
Theorem 8.2. A reduced finite rank torsion-free p-local group has semi-local
endomorphism ring. Therefore, it enjoys both the substitution and the cancellation
properties.
Proof. By Lemma 8.1, E=J is finite, so E is semi-local (cf. also Theorem 5.11 in
Chapter 16). Therefore, from Corollary 7.5 we deduce that A has the substitution,
and hence also the cancellation property.
ut
Power Cancellation A group A is said to have power cancellation if an
isomorphism A n Š C n for some n 2 implies A Š C:
In the next proof, we shall use notation and results from Sect. 12.
Proposition 8.3. Torsion-free p-local groups of finite rank have the power cancellation
property.
Proof. Suppose that the finite rank torsion-free groups A; C are p-local, and satisfy
A n D C n for some n 2. Evidently, A; C 2 add.G/. Denote by E the endomorphism
ring of G D A ˚ C. In the Arnold–Lady category equivalence (Theorem 12.2),