View source for Gamma-invariant in the theory of Abelian groups

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100502.png" />-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 [[#References|[a7]]]), such a group is free if it is of [[singular cardinal]]ity, so the invariant is defined for groups of [[regular cardinal]]ity (see [[Cardinal number|Cardinal number]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100503.png" /> is an Abelian group of regular uncountable cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100505.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100507.png" />-free if and only if every subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100508.png" /> of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g1100509.png" /> is free. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005010.png" /> can be written as the union of a continuous chain (called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005012.png" />-filtration) of free subgroups of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005013.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005014.png" />, where the continuity condition means that for every [[limit ordinal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005016.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005017.png" />-invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005018.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005019.png" /> or just <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005020.png" />, is defined to be the equivalence class, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005021.png" />, of

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005022.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005023.png" /> is defined to be the set of all subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005025.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005026.png" /> for some closed unbounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005028.png" />. (See [[Suslin hypothesis|Suslin hypothesis]] for the definitions of closed unbounded and stationary.) The equivalence class depends only on the isomorphism type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005029.png" /> and not on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005030.png" />-filtration, because any two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005031.png" />-filtrations agree on a closed unbounded subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005032.png" />; the equivalence classes of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005033.png" /> form a [[Boolean algebra|Boolean algebra]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005034.png" />, under the partial order induced by inclusion. The least element of this Boolean algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005035.png" />, is the class of all non-stationary subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005036.png" />. It can be proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005037.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005038.png" /> is free (see [[#References|[a1]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005040.png" />), every one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005041.png" /> members of the Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005042.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005043.png" />-invariant of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005044.png" />-free group of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005045.png" /> (see [[#References|[a6]]]). Assuming Gödel's [[axiom of constructibility]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005046.png" /> (see [[Gödel constructive set|Gödel constructive set]]), the same holds for all regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005047.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005048.png" />, of the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005049.png" /> can be given, assuming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005050.png" /> (see [[#References|[a6]]] and [[#References|[a5]]]).

Another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005051.png" />-invariant can be defined for use in connection with the [[Whitehead problem|Whitehead problem]] in Abelian group theory, and its generalizations. In this case, for any Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005054.png" /> is defined to be the equivalence class of

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005055.png" /></td> </tr></table>

when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005056.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005057.png" />-free of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005059.png" /> is written as the union, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005060.png" />, of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005061.png" />-filtration. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005062.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005063.png" />; the converse holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005064.png" /> of cardinality at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005065.png" />, assuming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005066.png" /> (see [[#References|[a2]]]).

Useful references for additional information are [[#References|[a3]]] and [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.C. Eklof,   "Methods of logic in abelian group theory" , ''Abelian Group Theory'' , ''Lecture Notes in Mathematics'' , '''616''' , Springer  (1977)  pp. 251–269</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.C. Eklof,   "Homological algebra and set theory"  ''Trans. Amer. Math. Soc.'' , '''227'''  (1977)  pp. 207–225</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.C. Eklof,   A.H. Mekler,   "Almost free modules" , North-Holland  (1990)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.C. Eklof,   A.H. Mekler,   S. Shelah,   "Almost disjoint abelian groups"  ''Israel J. Math.'' , '''49'''  (1984)  pp. 34–54</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.H. Mekler,   "How to construct almost free groups"  ''Canad. J. Math.'' , '''32'''  (1980)  pp. 1206–1228</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Shelah,   "A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals"  ''Israel J. Math.'' , '''21'''  (1975)  pp. 319–349</TD></TR></table>