Gamma-invariant in the theory of Abelian groups
-invariant
An invariant associated to an uncountable Abelian group and taking values in a Boolean algebra. Two groups with different invariants are non-isomorphic, but the converse fails in general: groups with the same invariant need not be isomorphic. The invariant is most commonly defined for almost-free groups (groups such that every subgroup of strictly smaller cardinality is a free Abelian group). By a theorem of S. Shelah (see [a7]), such a group is free if it is of singular cardinality, so the invariant is defined for groups of regular cardinality (see Cardinal number). If 
is an Abelian group of regular uncountable cardinality 
, 
is said to be 
-free if and only if every subgroup of 
of cardinality 
is free. In that case 
can be written as the union of a continuous chain (called a 
-filtration) of free subgroups of cardinality 
: 
, where the continuity condition means that for every limit ordinal 
, 
. The 
-invariant of 
, denoted by 
or just 
, is defined to be the equivalence class, 
, of
 |
is defined to be the set of all subsets 
of 
for which 
for some closed unbounded subset 
of 
. (See Suslin hypothesis for the definitions of closed unbounded and stationary.) The equivalence class depends only on the isomorphism type of 
and not on the choice of 
-filtration, because any two 
-filtrations agree on a closed unbounded subset of 
; the equivalence classes of subsets of 
form a Boolean algebra, 
, under the partial order induced by inclusion. The least element of this Boolean algebra, 
, is the class of all non-stationary subsets of 
. It can be proved that 
if and only if 
is free (see [a1]). For 
(
), every one of the 
members of the Boolean algebra 
is the 
-invariant of some 
-free group of cardinality 
(see [a6]). Assuming Gödel's axiom of constructibility, 
(see Gödel constructive set), the same holds for all regular 
which are not too large (e.g., less than the first inaccessible cardinal, or even the first Mahlo cardinal); in fact, a complete characterization, for any regular 
, of the range of 
can be given, assuming 
(see [a6] and [a5]).
Another 
-invariant can be defined for use in connection with the Whitehead problem in Abelian group theory, and its generalizations. In this case, for any Abelian groups 
and 
, 
is defined to be the equivalence class of
 |
when 
is 
-free of cardinality 
and 
is written as the union, 
, of a 
-filtration. Then 
implies 
; the converse holds for 
of cardinality at most 
, assuming 
(see [a2]).
Useful references for additional information are [a3] and [a4].
References
| [a1] | P.C. Eklof, "Methods of logic in abelian group theory" , Abelian Group Theory , Lecture Notes in Mathematics , 616 , Springer (1977) pp. 251–269 |
| [a2] | P.C. Eklof, "Homological algebra and set theory" Trans. Amer. Math. Soc. , 227 (1977) pp. 207–225 |
| [a3] | P.C. Eklof, "Set-theoretic methods: the uses of gamma invariants" , Abelian Groups , Lecture Notes in Pure and Appl. Math. , 146 , M. Dekker (1993) pp. 143–153 |
| [a4] | P.C. Eklof, A.H. Mekler, "Almost free modules" , North-Holland (1990) |
| [a5] | P.C. Eklof, A.H. Mekler, S. Shelah, "Almost disjoint abelian groups" Israel J. Math. , 49 (1984) pp. 34–54 |
| [a6] | A.H. Mekler, "How to construct almost free groups" Canad. J. Math. , 32 (1980) pp. 1206–1228 |
| [a7] | S. Shelah, "A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals" Israel J. Math. , 21 (1975) pp. 319–349 |
Gamma-invariant in the theory of Abelian groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gamma-invariant_in_the_theory_of_Abelian_groups&oldid=42426