# 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].

How to Cite This Entry:
Butler group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Butler_group&oldid=51314
This article was adapted from an original article by L. Fuchs (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article