Abelian Groups - László Fuchs [Springer]

550 14 Butler Groups
The next result deals with the relation between B 1 -andB 2 -groups in general. One
direction the implication is rather easy.
Theorem 5.3. B 2 -groups of any rank are B 1 -groups.
Proof. For a countable chain, the claim follows from Proposition 2.6. For transfinite
chains, we again refer to the proof of Lemma 4.1 in Chapter 9.
ut
(J) In the chain (14.4), the decent subgroups B are TEP-subgroups of the
B 2 -group B. This is an immediate consequence of Lemma 3.3 in view of the
preceding theorem.
Characterizations of B 2 -Groups The next result establishes a larger supply of
decent subgroups in B 2 -groups.
Theorem 5.4 (Albrecht–Hill [1]). A torsion-free group G is a B 2 -group if and only
if it admits an H.@ 0 /-family B of decent subgroups. Equivalently, it has an H.@ 0 /-
family of decent TEP-subgroups.
Furthermore, if A; B 2 B and B < A, then A=BisaB 2 -group.
Proof. One way the claim is obvious, since from an H.@ 0 /-family of decent
subgroups we can extract a smooth chain with countable factors which are
B 2 -groups. Such a chain can be refined to a desired chain with finite rank factors.
For the converse, assume that G is a B 2 -group. We appeal to Theorem 5.5 in
Chapter 1 to obtain an H.@ 0 /-family B, and to its proof to argue that, for blocked
subsets S, the subgroups G.S/ are decent and TEP-subgroups in G. SoletC=G.S/
be a finite rank pure subgroup of G=G.S/. There is a finite blocked subset S 0 that
contains representatives of a maximal independent set in C=G.S/.ThenC D G.S/C
.C\G.S 0 //, where the intersection is a pure subgroup of G.S 0 /, so a finite rank Butler
group. Thus G.S/ is decent in G. It is also a TEP-subgroup, because by creating an
increasing sequence of blocked subgroups, we can find a chain like (14.4) passing
through G.S/ and climbing up to G; finally, we refer to (D).
The last argument implies that all the factors in B are B 2 -groups.
ut
Equipped with this theorem, we can derive a not so obvious corollary (that we
could not claim in (C)).
Corollary 5.5. Summands of B 2 -groups are B 2 -groups.
Proof. Let A D G ˚ H be a direct decomposition of a B 2 -group, and B an H.@ 0 /-
family of decent subgroups of A. Define the subfamily
B ? DfB 2 B j B D B \ G ˚ B \ Hg:
In view of Lemma 5.4 in Chapter 1,thisisanH.@ 0 /-family of decent subgroups of
A,andthesetfB \ G j B 2 B ? g is an H.@ 0 /-family of decent subgroups of G. ut
It might help the reader to navigate a passage through the theory of general Butler groups,
through this rich and fascinating subject, if we explain briefly our approach and our limited
goal, before getting involved more sophisticated arguments. A principal aim of the theory is to