Abelian Groups - László Fuchs [Springer]

1 Characteristic and Type: Finite Rank Groups 413
Lemma 1.4. Let R and S be rational groups of types t and s, respectively. Then
(i) Hom.R; S/ D 0 unless t s;
(ii) if t s,thenHom.R; S/ is a rational group of type s W t;
(iii) Hom.R; R/ is a rational group of idempotent type t 0 D t W t; and
(iv) Hom.Hom.R; R/; R/ is a rational group of type t.
ut
Finite Rank Groups A torsion-free group A of finite rank n is a subgroup
of an n-dimensional Q-vector space V, so we can describe A by starting with an
independent set a 1 ;:::;a n , and then adjoining more elements of V as generators of
A, responsible for divisibility properties.
Example 1.5. Let P be a set of primes, and p; q primes … P.Then
A Dhp 2 a; q 1 b; P 1 .a C b/i
is the group of rank 2 in which a; b are independent elements, and the additional generators are:
p 2 a; q n b for every n 2 N,andr 1 .a C b/ for every r 2 P.
Type.A/ will denote the Typeset of A, i.e. the set of types of the non-zero
elements of A. Type.A/ can be infinite even if rk.A/ is finite. We also talk of
Cotype.A/ as the set of cotypes, i.e. types of rank one torsion-free factor groups of
A; but this gives useful information only if A is of finite rank. IT.A/ and OT.A/ are
the notations for the inner and outer types of A:IT.A/ is defined as the intersection
of all types t.a/ for all a 2 A, andOT.A/ is the union of all types of rank 1 torsionfree
factor groups of A.Wehave:
Lemma 1.6. Let A be a torsion-free group of finite rank, a 1 ;:::;a n a
maximal independent set in A, and A i the pure subgroup of A generated by
a 1 ;:::;a i 1 ; a iC1 ;:::;a n .Then
IT.A/ D t.a 1 / ^ :::^ t.a n / and OT.A/ D t.A=A 1 /_ :::_t.A=A n /:
Proof. What we have to show for IT.A/ is that, for every a 2 A, the type t.a/ is
larger than or equal to the stated intersection. If we write a D r 1 a 1 CCr n a n
with r i 2 Q, then it is clear that we must have t.a/ t.r 1 a 1 / ^ ::: ^ t.r n a n / D
t.a 1 / ^ :::^ t.a n /:
For OT.A/ we argue that, since A 1 \ :::\ A n D 0, there is an embedding A !
A=A 1 ˚˚A=A n in a direct sum of rank 1 groups, call it NA for a moment. By
definition, _ i t.A=A i / OT.A/. On the other hand, it is obvious that OT.A/
OT. NA/. Since every homomorphism of a direct sum of rank 1 groups into Q carries
each rank 1 summand either to 0 or isomorphically into Q, it is evident that OT. NA/ D
_ i t.A=A i /.
ut
Example 1.7. For a finite rank A, wehaveIT.A/ =OT.A/ if and only if A is a direct sum of
isomorphic rank 1 groups. The ‘if’ part is pretty obvious, while the converse is a consequence of
Lemma 3.6.