Butler group

A torsion-free Abelian group of finite rank (cf. Rank of a group) that is a pure subgroup of a completely decomposable group of finite rank. Equivalently, a Butler group is an epimorphic image of a completely decomposable group of finite rank [a5].

Clearly, all completely decomposable Abelian groups of finite rank are Butler groups, and so are their extensions by finite groups. There are lots of other examples of Butler groups. Let $ A $ be a direct sum, $ A = A _ {1} \oplus A _ {2} \oplus A _ {3} $, where the $ A _ {i} $ are rank-one torsion-free groups such that the elements $ a _ {i} \in A _ {i} $ have characteristics $ ( \infty, \infty, 0, 0, \dots ) $, $ ( \infty, 0, \infty, 0, 0, \dots ) $ and $ ( 0, \infty, \infty, 0, 0, \dots ) $, respectively. The pure subgroup $ B $ of $ A $ generated by the elements $ a _ {1} - a _ {2} $, $ a _ {2} - a _ {3} $, $ a _ {3} - a _ {1} $ is a rank-two indecomposable Butler group. The class of Butler groups is closed under the formation of finite direct sums, pure subgroups and torsion-free epimorphic images. The type-set (i.e., the set of types of the non-zero elements) of a Butler group is always finite.

There are various other characterizations of Butler groups.

i) The following properties characterize Butler groups $ B $ among the finite-rank torsion-free groups [a5]: a) the type-set of $ B $ is finite; b) for each type $ t $, the subgroup $ B ^ {*} ( t ) $ generated by the elements of type $ > t $ in $ B $ has finite index in its purification $ B ^ {*} ( t ) * $; c) for each type $ t $, $ B ( t ) = B _ {t} \oplus B ^ {*} ( t ) * $, where $ B ( t ) $ is the set of elements of type $ \geq t $ in $ B $ and $ B _ {t} $ is a homogeneous completely decomposable group of type $ t $.

ii) A finite-rank torsion-free group $ B $ is a Butler group if and only if there is a partition $ \Pi = \Pi _ {1} \cup \dots \cup \Pi _ {k} $ of the set $ \Pi $ of prime numbers such that for each $ i $( $ i = 1 \dots k $), the tensor product $ B \otimes \mathbf Z _ {i} $ is a completely decomposable group with totally ordered type-set (here, $ \mathbf Z _ {i} $ denotes the localization of $ \mathbf Z $ at the set $ \Pi _ {i} $ of primes) [a3].

iii) A finite-rank torsion-free group $ B $ is Butler exactly if it satisfies $ { \mathop{\rm Bext} } ^ {1} ( B,T ) = 0 $ for all torsion Abelian groups $ T $[a4]. Here, $ { \mathop{\rm Bext} } ^ {1} $ denotes the group of equivalence classes of extensions of $ T $ by $ B $ in which $ T $ is a balanced subgroup.

The classification of Butler groups has not gotten too far (1996). Two important classes have been characterized by invariants up to quasi-isomorphism. These are the Butler groups of Richman type [a13] and their duals. (A Butler group $ B $ is of Richman type if it is a corank-one pure subgroup in a completely decomposable group of finite rank. See [a2], [a10], [a11].)

It is worthwhile mentioning that there is a close connection between Butler groups and representations of finite partially ordered sets.

Butler groups $ B $ of countable rank were introduced in [a4]. Of the numerous equivalent characterizations, the following are noteworthy:

i) $ { \mathop{\rm Bext} } ^ {1} ( B,T ) = 0 $ for all torsion Abelian groups $ T $;

ii) $ B $ is the union of an ascending chain of (finite-rank) Butler subgroups which are pure in $ B $;

iii) every finite-rank pure subgroup of $ B $ is a Butler group.

The study of Butler groups of large cardinalities often requires additional set-theoretical hypotheses beyond the axioms of ZFC (cf. Set theory). There are two kinds of Butler groups of arbitrary cardinality [a4]: $ B $ is a $ B _ {1} $- group if $ { \mathop{\rm Bext} } ^ {1} ( B,T ) = 0 $ for all torsion Abelian groups $ T $, and a $ B _ {2} $- group if it is the union of a continuous well-ordered ascending chain of pure subgroups $ B _ \alpha $ such that, for all $ \alpha $, $ B _ {\alpha + 1 } = B _ \alpha + G _ \alpha $ for some finite-rank Butler group $ G _ \alpha $. All $ B _ {2} $- groups are $ B _ {1} $- groups, and the converse is one of the major open problems in Abelian group theory. It is known that the continuum hypothesis, CH, guarantees that all $ B _ {1} $- groups of cardinality $ \leq \aleph _ \omega $ are $ B _ {2} $- groups [a6], while in Gödel's constructible universe $ L $, the same holds without cardinality restrictions [a9]. A useful criterion is: assuming CH, a $ B _ {1} $- group $ B $ is a $ B _ {2} $- group if and only if $ { \mathop{\rm Bext} } ^ {2} ( B,T ) = 0 $ for all torsion groups $ T $[a12].

The other important problem is to find conditions under which a pure subgroup $ A $ of a $ B _ {2} $- group $ B $ is likewise a $ B _ {2} $- group. A necessary and sufficient condition is the existence of a continuous well-ordered ascending chain of $ B _ {2} $- subgroups from $ A $ to $ B $ with rank- $ 1 $ factors [a8]. A related problem is whether or not $ { \mathop{\rm Bext} } ^ {2} ( G,T ) = 0 $ for all torsion-free groups $ G $ and all torsion groups $ T $. In [a7] it is shown that CH is a necessary condition for the vanishing of $ { \mathop{\rm Bext} } ^ {2} $, while in [a9] it is proved that the hypothesis $ V = L $ is a sufficient condition. It should be pointed out that $ { \mathop{\rm Bext} } ^ {3} ( G,T ) $ always vanishes, provided CH is assumed [a1]; more generally, $ { \mathop{\rm Bext} } ^ {n + 2 } ( G,T ) $ vanishes if $ \aleph _ {n} $ is the continuum for some integer $ n \geq 1 $[a8]. Another useful result, valid in ZFC, states that in a balanced-projective resolution $ 0 \rightarrow K \rightarrow C \rightarrow B \rightarrow 0 $ of a $ B _ {1} $- group $ B $( i.e., $ C $ is completely decomposable and $ K $ is balanced in $ C $), if one of $ B $, $ K $ is a $ B _ {2} $- group, then so is the other [a8].

References

[a1]	U. Albrecht, P. Hill, "Butler groups of infinite rank and Axiom 3" Czechosl. Math. J. , 37 (1987) pp. 293–309
[a2]	D. Arnold, C. Vinsonhaler, "Invariants for a class of torsion-free abelian groups" Proc. Amer. Math. Soc. , 105 (1989) pp. 293–300
[a3]	L. Bican, "Purely finitely generated abelian groups" Comment. Math. Univ. Carolin. , 21 (1980) pp. 209–218
[a4]	L. Bican, L. Salce, "Butler groups of infinite rank" , Abelian Group Theory , Lecture Notes in Mathematics , 1006 , Springer (1983) pp. 171–189
[a5]	M.C.R. Butler, "A class of torsion-free abelian groups of finite rank" Proc. London Math. Soc. , 15 (1965) pp. 680–698
[a6]	M. Dugas, P. Hill, K.M Rangaswamy, "Infinite rank Butler groups II" Trans. Amer. Math. Soc. , 320 (1990) pp. 643–664
[a7]	M. Dugas, B. Thomé, "The functor Bext and the negation of CH" Forum Math. , 3 (1991) pp. 23–33
[a8]	L. Fuchs, "Butler groups of infinite rank" J. Pure Appl. Algebra , 98 (1995) pp. 25–44
[a9]	L. Fuchs, M. Magidor, "Butler groups of arbitrary cardinality" Israel J. Math. , 84 (1993) pp. 239–263
[a10]	L. Fuchs, C. Metelli, "On a class of Butler groups" Manuscr. Math. , 71 (1991) pp. 1–28
[a11]	P. Hill, C. Megibben, "The classification of certain Butler groups" J. Algebra , 160 (1993) pp. 524–551
[a12]	K.M. Rangaswamy, "A homological characterization of Butler groups" Proc. Amer. Math. Soc. , 121 (1994) pp. 409–415
[a13]	F. Richman, "An extension of the theory of completely decomposable torsion-free abelian groups" Trans. Amer. Math. Soc. , 279 (1983) pp. 175–185