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 be a direct sum, , where the are rank-one torsion-free groups such that the elements have characteristics , and , respectively. The pure subgroup of generated by the elements , , 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 among the finite-rank torsion-free groups [a5]: a) the type-set of is finite; b) for each type , the subgroup generated by the elements of type in has finite index in its purification ; c) for each type , , where is the set of elements of type in and is a homogeneous completely decomposable group of type .

ii) A finite-rank torsion-free group is a Butler group if and only if there is a partition of the set of prime numbers such that for each (), the tensor product is a completely decomposable group with totally ordered type-set (here, denotes the localization of at the set of primes) [a3].

iii) A finite-rank torsion-free group is Butler exactly if it satisfies for all torsion Abelian groups [a4]. Here, denotes the group of equivalence classes of extensions of by in which 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 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 of countable rank were introduced in [a4]. Of the numerous equivalent characterizations, the following are noteworthy:

i) for all torsion Abelian groups ;

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

iii) every finite-rank pure subgroup of 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]: is a -group if for all torsion Abelian groups , and a -group if it is the union of a continuous well-ordered ascending chain of pure subgroups such that, for all , for some finite-rank Butler group . All -groups are -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 -groups of cardinality are -groups [a6], while in Gödel's constructible universe (cf. also Gödel constructive set), the same holds without cardinality restrictions [a9]. A useful criterion is: assuming CH, a -group is a -group if and only if for all torsion groups [a12].

The other important problem is to find conditions under which a pure subgroup of a -group is likewise a -group. A necessary and sufficient condition is the existence of a continuous well-ordered ascending chain of -subgroups from to with rank- factors [a8]. A related problem is whether or not for all torsion-free groups and all torsion groups . In [a7] it is shown that CH is a necessary condition for the vanishing of , while in [a9] it is proved that the hypothesis is a sufficient condition. It should be pointed out that always vanishes, provided CH is assumed [a1]; more generally, vanishes if is the continuum for some integer [a8]. Another useful result, valid in ZFC, states that in a balanced-projective resolution of a -group (i.e., is completely decomposable and is balanced in ), if one of , is a -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