Abelian Groups - László Fuchs [Springer]

5 Vector Groups 511
Isomorphism of Vector Groups At this point the natural question is: when are
two vector groups isomorphic? To answer this question, we prove the following
theorem. We denote by t 0 the largest idempotent type with t 0 t,i.e.t 0 D t W t.
Theorem 5.5. Suppose
V D Y t V t and W D Y t W t
are vector groups where V t and W t are elementary vector groups of type t, and t
runs over the different types.
(i) (Sa¸siada [3]) V Š W if and only if V t Š W t for every type t.
(ii) If V t D Q i2I R i Š W t D Q j2J R j with a type t 6Š t.Q/, thenjIj DjJj whenever
I or J is non-measurable.
Proof.
(i) The “if” part being obvious, assume there is an isomorphism W V ! W. Let
t W V ! V t and t W W ! W t denote the coordinate projections in the two
direct products. By Lemma 5.4, every homomorphism of V t into W s is trivial
unless the types satisfy s t. Therefore, V t Q st W s.Ifv 2 V t ,then
v D w C w 0 with w 2 W t and w 0 2 Q s>t W s: AgainbyLemma5.4, the
last group has only the trivial homomorphisms into V t and into W t , whence
t 1 w 0 D 0 D t w 0 follows. This shows that v D t v D t 1 w and w D
t w D t v; consequently, the map t 1 t is the identity on V t . Changing
the roles of V t and W t , the stated isomorphism is immediate.
(ii) Let R and R 0 be rational groups of types t and t 0 , respectively. In view
of Corollary 2.10, wehaveHom.V t ; R/ Š ˚i2I Hom.R i ; R/ D ˚i2I R 0 and
Hom.W t ; R/ Š ˚j2J R 0 . Both are completely decomposable groups whence
V t Š W t implies jIj D jJj. ((ii) holds also for measurable index sets,
see Theorem 6.1 below.)
ut
Summands of Non-measurable Vector Groups One of the most interesting
facts on vector groups is concerned with direct summands in the non-measurable
case.
Theorem 5.6 (O’Neill [2]).
(i) Summands of reduced vector groups are direct products of summands of
elementary vector groups of different types.
(ii) (Balcerzyk–Bialynicki-Birula–Łoś[1])Summands of reduced non-measurable
vector groups are again vector groups.
Proof.
(i) Let V D Q t V t with elementary vector groups V t of type t. The projections in
this decomposition will be denoted by t . For each type t, the subgroups
V t D Y st
V s and V t D Y s>t
V s
are fully invariant, and satisfy V t D V t ˚ V t .