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- W1995757051 abstract "Recently R. G. Swan [51 has proven the following theorem with no commutative assumptions whatsoever: If a torsion free group G contains a free subgroup of finite index, then G is free. Of course, if G is abelian, there is a trivial generalization to the case where G/F is assumed to be bounded with F free (that is, n(G/F) = 0 where n$O0GZ). However, in this note we show that Swan's theorem has an even more striking improvement than the one suggested above for G an abelian group. For example, G/F may be a subgroup of a totally projective p-group of arbitrary length. In the ensuing discussion all groups will be abelian and p will be an arbitrary but fixed prime. We denote the group of integers by Z and the group of rational numbers by Q. A cotorsion functor S (defined by Nunke in [3]) is one that has a representing sequence Z>-+M--->H where H is torsion such that, for a group G, SG= Image(Hom(M, G)--Hom(Z, G)_G) and if f:G--A then S( =f SG. We further use the notation Sp when H is a p-group (that is, Sp is a p-coprimary functor in the sense of Nunke) and call Sp reduced if H is reduced. Hill and Megibben [2] call a group G a weak S-projective if for any free resolution Fo >-4F---> G of G and any group X we have that 6x(S Hom(Fo, X)) =0 where" @default.
- W1995757051 created "2016-06-24" @default.
- W1995757051 creator A5011278013 @default.
- W1995757051 date "1970-04-01" @default.
- W1995757051 modified "2023-09-24" @default.
- W1995757051 title "Extensions of free groups by torsion groups" @default.
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- W1995757051 doi "https://doi.org/10.1090/s0002-9939-1970-0257228-4" @default.
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