Abelian Groups - László Fuchs [Springer]

8 p -Purity 385
then B has an H./-family of -separative subgroups if and only if C has an H. C /-family of
C -separative subgroups. It was also proved that the balanced-projective dimension of p-groups
can be any integer 0 or 1; in particular, for each n 0,anyp !n -high subgroup of a p-group A
of cardinality @ n satisfying p !n A ¤ 0 has balanced-projective dimension exactly n. (A corollary is
that balanced subgroups in totally projective groups need not be totally projective.)
Hill [19] proved a strong isomorphism theorem on balanced subgroups of totally projective p-
groups: If B; B 0 are balanced subgroups in the totally projective p-groups A; A 0 such that they have
the same UK-invariants and satisfy A=B Š A 0 =B 0 , then there is an isomorphism A ! A 0 carrying
B onto B 0 . Warfield [4] proved a criterion on the isomorphism of two isotype, -dense subgroups,
C and C 0 , of a totally projective p-group G of limit length . IfdimGŒp=CŒp D dim GŒp=C 0 Œp,
then there is an automorphism of G inducing an isomorphism C Š C 0 .
Relative balanced-projective resolutions were introduced by Bican–Fuchs [Comm. Algebra 22,
1031–1036 (1994)] for modules. They will play a more substantial role in Chapter 14.
Some classes of isotype subgroups of totally projective p-groups have well-developed theories.
These include the classes of S-groups and A-groups. Let denote a limit ordinal not cofinal with
!. An isotype subgroup H of a totally projective p-group G is called a -elementary S-group
if G=H Š Z.p 1 / and G D H C p G for all < . H is a -elementary A-group if G=H
is (possibly non-reduced) totally projective and p .G=H/ D .H C p G/=H for all < .An
S-group (A-group) is the direct sum of -elementary S-groups (A-groups) for various ordinals .
S-groups can also be defined as torsion subgroups of Warfield groups. To classify S-andA-groups
(within the mentioned class, of course), one needs one invariant in addition to the UK-invariants.
The proofs are long and difficult. See Warfield [6], Hill [20], Hill–Megibben [6], as well as the
literature quoted there.
Exercises
(1) Show that compatibility is reflexive, but not transitive.
(2) If AkB and A\B D 0,thenh.a Cb/ D minfh.a/; h.b/g for all a 2 A and b 2 B.
(3) There exists a separative chain from A to G if there is a smooth chain A D A 0 <
< A˛ < < A D G of separative subgroups such that the cardinalities of
the factor groups A˛C1 =A˛ .˛ < / are @ 1 .
(4) Let C be a subgroup of the p-group A such that nA C A for some n 2 N.
Then C is a direct sum of countable groups if and only if so is A.
(5) (Hill) Let A be a direct sum of countable groups, and suppose that the UKinvariants
of A are @ 1 . An isotype subgroup of A is a direct sum of countable
groups if and only if it is separative in A.
(6) If A is totally projective, then every endomorphism of p A extends to an
endomorphism of A.
8 p -Purity
There exist several versions of generalized purity, each of which is motivated by a
particular feature of pure subgroups. In this section, we deal with one that depends
on an arbitrarily chosen ordinal, and seems to be of special interest from the
homological point of view.