Abelian Groups - László Fuchs [Springer]

152 5 Purity and Basic Subgroups
Proof. Let B be a subgroup of A of cardinality . Consider all equations nx D b for
all n 2 N and all b 2 B that are solvable in x 2 A. For each such equation, we adjoin
a solution a n;b 2 A to B in order to obtain a subgroup B 1 A in which all these
equations are solvable; thus, B 1 DhB; a n;b .8 n; b/i. We repeat this precess with B 1
in place of B to get a subgroup B 2 in which all equations with right members in
B 1 are solvable whenever they admit a solution in A. Thus proceeding, we form the
union G of the chain B B 1 B m ::: .m m, then
p 4n 1 a 2n C p 4n .a 2nC1 c 2nC1 / 2 p m CŒp LŒp. It follows that p 4n 1 a 2n 2 LŒp which is clearly
impossible. Thus no such C exists.
A subgroup C of A is purifiable if there exists a pure subgroup G A containing
C such that A has no pure subgroup containing C and properly contained in G.There
is no satisfactory characterization of purifiable subgroups. We state a relevant result
without proof.
Theorem 1.7 (Hill–Megibben [1]). A p-group A has the property that all of its
subgroups are purifiable if and only if A D B ˚ D, where B is a bounded and D is a
divisible group.
ut
It is useful to keep in mind that groups admit pure @ 0 -filtrations:
Proposition 1.8. For every pure subgroup G of A, there is a smooth chain G D
G 0 < G 1 < < G < < G D A of pure subgroups for some ordinal such
that each G C1 =G is countable.
Proof. To obtain G C1 from G , choose G C1 =G as a countable pure subgroup of
A=G (Theorem 1.5).
ut
A word on purity in p-adic modules. Since p-adic modules are q-divisible for
every prime q ¤ p, purity is the same as p-purity. (p-purity in the p-adic sense is the
same as in groups.)
Neat Subgroups There exist several concepts related to purity that deserve
mentioning, besides the isotype subgroups that will be discussed in Chapters 11
and 15.