Namespaces
Variants
Actions

Difference between revisions of "Butler group"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
b1110801.png
 +
$#A+1 = 95 n = 0
 +
$#C+1 = 95 : ~/encyclopedia/old_files/data/B111/B.1101080 Butler group
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
A torsion-free [[Abelian group|Abelian group]] of finite rank (cf. [[Rank of a group|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 [[#References|[a5]]].
 
A torsion-free [[Abelian group|Abelian group]] of finite rank (cf. [[Rank of a group|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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b1110801.png" /> be a direct sum, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b1110802.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b1110803.png" /> are rank-one torsion-free groups such that the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b1110804.png" /> have characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b1110805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b1110806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b1110807.png" />, respectively. The pure subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b1110808.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b1110809.png" /> generated by the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108012.png" /> 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.
+
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.
 
There are various other characterizations of Butler groups.
  
i) The following properties characterize Butler groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108013.png" /> among the finite-rank torsion-free groups [[#References|[a5]]]: a) the type-set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108014.png" /> is finite; b) for each type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108015.png" />, the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108016.png" /> generated by the elements of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108018.png" /> has finite index in its purification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108019.png" />; c) for each type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108022.png" /> is the set of elements of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108025.png" /> is a homogeneous completely decomposable group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108026.png" />.
+
i) The following properties characterize Butler groups $  B $
 +
among the finite-rank torsion-free groups [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108027.png" /> is a Butler group if and only if there is a partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108028.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108029.png" /> of prime numbers such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108031.png" />), the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108032.png" /> is a completely decomposable group with totally ordered type-set (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108033.png" /> denotes the localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108034.png" /> at the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108035.png" /> of primes) [[#References|[a3]]].
+
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) [[#References|[a3]]].
  
iii) A finite-rank torsion-free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108036.png" /> is Butler exactly if it satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108037.png" /> for all torsion Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108038.png" /> [[#References|[a4]]]. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108039.png" /> denotes the group of equivalence classes of extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108040.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108041.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108042.png" /> is a balanced subgroup.
+
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 $[[#References|[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 [[#References|[a13]]] and their duals. (A Butler group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108043.png" /> is of Richman type if it is a corank-one pure subgroup in a completely decomposable group of finite rank. See [[#References|[a2]]], [[#References|[a10]]], [[#References|[a11]]].)
+
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 [[#References|[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 [[#References|[a2]]], [[#References|[a10]]], [[#References|[a11]]].)
  
 
It is worthwhile mentioning that there is a close connection between Butler groups and representations of finite partially ordered sets.
 
It is worthwhile mentioning that there is a close connection between Butler groups and representations of finite partially ordered sets.
  
Butler groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108044.png" /> of countable rank were introduced in [[#References|[a4]]]. Of the numerous equivalent characterizations, the following are noteworthy:
+
Butler groups $  B $
 +
of countable rank were introduced in [[#References|[a4]]]. Of the numerous equivalent characterizations, the following are noteworthy:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108045.png" /> for all torsion Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108046.png" />;
+
i) $  { \mathop{\rm Bext} }  ^ {1} ( B,T ) = 0 $
 +
for all torsion Abelian groups $  T $;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108047.png" /> is the union of an ascending chain of (finite-rank) Butler subgroups which are pure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108048.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108049.png" /> is a Butler group.
+
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|Set theory]]). There are two kinds of Butler groups of arbitrary cardinality [[#References|[a4]]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108050.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108052.png" />-group if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108053.png" /> for all torsion Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108054.png" />, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108056.png" />-group if it is the union of a continuous well-ordered ascending chain of pure subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108057.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108059.png" /> for some finite-rank Butler group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108060.png" />. All <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108061.png" />-groups are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108062.png" />-groups, and the converse is one of the major open problems in Abelian group theory. It is known that the [[Continuum hypothesis|continuum hypothesis]], CH, guarantees that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108063.png" />-groups of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108064.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108065.png" />-groups [[#References|[a6]]], while in Gödel's constructible universe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108066.png" /> (cf. also [[Gödel constructive set|Gödel constructive set]]), the same holds without cardinality restrictions [[#References|[a9]]]. A useful criterion is: assuming CH, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108067.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108068.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108069.png" />-group if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108070.png" /> for all torsion groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108071.png" /> [[#References|[a12]]].
+
The study of Butler groups of large cardinalities often requires additional set-theoretical hypotheses beyond the axioms of ZFC (cf. [[Set theory|Set theory]]). There are two kinds of Butler groups of arbitrary cardinality [[#References|[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|continuum hypothesis]], CH, guarantees that all $  B _ {1} $-
 +
groups of cardinality $  \leq  \aleph _  \omega  $
 +
are $  B _ {2} $-
 +
groups [[#References|[a6]]], while in Gödel's constructible universe $  L $(
 +
cf. also [[Gödel constructive set|Gödel constructive set]]), the same holds without cardinality restrictions [[#References|[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 $[[#References|[a12]]].
  
The other important problem is to find conditions under which a [[Pure subgroup|pure subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108072.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108073.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108074.png" /> is likewise a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108075.png" />-group. A necessary and sufficient condition is the existence of a continuous well-ordered ascending chain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108076.png" />-subgroups from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108077.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108078.png" /> with rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108079.png" /> factors [[#References|[a8]]]. A related problem is whether or not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108080.png" /> for all torsion-free groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108081.png" /> and all torsion groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108082.png" />. In [[#References|[a7]]] it is shown that CH is a necessary condition for the vanishing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108083.png" />, while in [[#References|[a9]]] it is proved that the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108084.png" /> is a sufficient condition. It should be pointed out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108085.png" /> always vanishes, provided CH is assumed [[#References|[a1]]]; more generally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108086.png" /> vanishes if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108087.png" /> is the continuum for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108088.png" /> [[#References|[a8]]]. Another useful result, valid in ZFC, states that in a balanced-projective resolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108089.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108090.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108091.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108092.png" /> is completely decomposable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108093.png" /> is balanced in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108094.png" />), if one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108096.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111080/b11108097.png" />-group, then so is the other [[#References|[a8]]].
+
The other important problem is to find conditions under which a [[Pure subgroup|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 [[#References|[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 [[#References|[a7]]] it is shown that CH is a necessary condition for the vanishing of $  { \mathop{\rm Bext} }  ^ {2} $,  
 +
while in [[#References|[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 [[#References|[a1]]]; more generally, $  { \mathop{\rm Bext} } ^ {n + 2 } ( G,T ) $
 +
vanishes if $  \aleph _ {n} $
 +
is the continuum for some integer $  n \geq  1 $[[#References|[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 [[#References|[a8]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U. Albrecht,  P. Hill,  "Butler groups of infinite rank and Axiom 3"  ''Czechosl. Math. J.'' , '''37'''  (1987)  pp. 293–309</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Arnold,  C. Vinsonhaler,  "Invariants for a class of torsion-free abelian groups"  ''Proc. Amer. Math. Soc.'' , '''105'''  (1989)  pp. 293–300</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Bican,  "Purely finitely generated abelian groups"  ''Comment. Math. Univ. Carolin.'' , '''21'''  (1980)  pp. 209–218</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Bican,  L. Salce,  "Butler groups of infinite rank" , ''Abelian Group Theory'' , ''Lecture Notes in Mathematics'' , '''1006''' , Springer  (1983)  pp. 171–189</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.C.R. Butler,  "A class of torsion-free abelian groups of finite rank"  ''Proc. London Math. Soc.'' , '''15'''  (1965)  pp. 680–698</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Dugas,  P. Hill,  K.M Rangaswamy,  "Infinite rank Butler groups II"  ''Trans. Amer. Math. Soc.'' , '''320'''  (1990)  pp. 643–664</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Dugas,  B. Thomé,  "The functor Bext and the negation of CH"  ''Forum Math.'' , '''3'''  (1991)  pp. 23–33</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Fuchs,  "Butler groups of infinite rank"  ''J. Pure Appl. Algebra'' , '''98'''  (1995)  pp. 25–44</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  L. Fuchs,  M. Magidor,  "Butler groups of arbitrary cardinality"  ''Israel J. Math.'' , '''84'''  (1993)  pp. 239–263</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  L. Fuchs,  C. Metelli,  "On a class of Butler groups"  ''Manuscr. Math.'' , '''71'''  (1991)  pp. 1–28</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  P. Hill,  C. Megibben,  "The classification of certain Butler groups"  ''J. Algebra'' , '''160'''  (1993)  pp. 524–551</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  K.M. Rangaswamy,  "A homological characterization of Butler groups"  ''Proc. Amer. Math. Soc.'' , '''121'''  (1994)  pp. 409–415</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  F. Richman,  "An extension of the theory of completely decomposable torsion-free abelian groups"  ''Trans. Amer. Math. Soc.'' , '''279'''  (1983)  pp. 175–185</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U. Albrecht,  P. Hill,  "Butler groups of infinite rank and Axiom 3"  ''Czechosl. Math. J.'' , '''37'''  (1987)  pp. 293–309</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Arnold,  C. Vinsonhaler,  "Invariants for a class of torsion-free abelian groups"  ''Proc. Amer. Math. Soc.'' , '''105'''  (1989)  pp. 293–300</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Bican,  "Purely finitely generated abelian groups"  ''Comment. Math. Univ. Carolin.'' , '''21'''  (1980)  pp. 209–218</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Bican,  L. Salce,  "Butler groups of infinite rank" , ''Abelian Group Theory'' , ''Lecture Notes in Mathematics'' , '''1006''' , Springer  (1983)  pp. 171–189</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.C.R. Butler,  "A class of torsion-free abelian groups of finite rank"  ''Proc. London Math. Soc.'' , '''15'''  (1965)  pp. 680–698</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Dugas,  P. Hill,  K.M Rangaswamy,  "Infinite rank Butler groups II"  ''Trans. Amer. Math. Soc.'' , '''320'''  (1990)  pp. 643–664</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Dugas,  B. Thomé,  "The functor Bext and the negation of CH"  ''Forum Math.'' , '''3'''  (1991)  pp. 23–33</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Fuchs,  "Butler groups of infinite rank"  ''J. Pure Appl. Algebra'' , '''98'''  (1995)  pp. 25–44</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  L. Fuchs,  M. Magidor,  "Butler groups of arbitrary cardinality"  ''Israel J. Math.'' , '''84'''  (1993)  pp. 239–263</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  L. Fuchs,  C. Metelli,  "On a class of Butler groups"  ''Manuscr. Math.'' , '''71'''  (1991)  pp. 1–28</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  P. Hill,  C. Megibben,  "The classification of certain Butler groups"  ''J. Algebra'' , '''160'''  (1993)  pp. 524–551</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  K.M. Rangaswamy,  "A homological characterization of Butler groups"  ''Proc. Amer. Math. Soc.'' , '''121'''  (1994)  pp. 409–415</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  F. Richman,  "An extension of the theory of completely decomposable torsion-free abelian groups"  ''Trans. Amer. Math. Soc.'' , '''279'''  (1983)  pp. 175–185</TD></TR></table>

Revision as of 06:29, 30 May 2020


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 $( cf. also Gödel constructive set), 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
How to Cite This Entry:
Butler group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Butler_group&oldid=46178
This article was adapted from an original article by L. Fuchs (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article