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 free; see 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