Abelian Groups - László Fuchs [Springer]

3 Pure-Exact Sequences 159
(6) (Cutler, E. Walker) If the groups A and G satisfy nA Š nG for some n 2 N,
then there exist groups A 0 and G 0 such that A ˚ A 0 Š G ˚ G 0 with nA 0 D 0 D
nG 0 . [Hint: find maximal n-bounded subgroups.]
(7) If nA D˚i2I C i is a direct decomposition, then there exist subgroups B i such
that A D˚i2I B i with nB i D C i .
(8) If B is a pure subgroup of A,then.A=B/Œn Š AŒn=BŒn for every n 2 N.
(9) (Kulikov) Suppose a 2 A is an element of smallest finite order in the coset
a C pA.Thenhai is a summand of A.
(10) (Mader) Let A be an infinite p-group such that jA=p n Aj < jAj for some n 2 N.
Then A contains a p n -bounded summand of cardinality jAj.
(11) The closure C of a pure subgroup C of A (in the Z-adic topology) is pure if
and only if .A=C/ 1 is a divisible group.
(12) Let B ˚ C be a pure subgroup in a reduced torsion-free group A. Then the
closures B ; C in the Z-adic topology of A are still disjoint, and B ˚ C is
pure in A.
(13) (Göbel–Goldsmith) In every group ¤ 0, the set of all proper pure subgroups
contains a maximal member. [Hint: argue separately for torsion-free groups.]
(14) A group is called absolutely pure if it is a pure subgroup in any group in
which it is contained as a subgroup. Show that D is absolutely pure if and only
if it is divisible.
(15) A subgroup B is -pure in A if and only if every system of equations over B
with less than unknowns is solvable in B whenever it admits a solution in A.
[Hint: Theorem 2.12.]
(16) A subgroup G of a group A can be embedded in an @ -pure subgroup of
cardinality jGj @
,where D 1 or according as is a successor
or a limit ordinal. [Hint: argue as in Theorem 1.5 and preceding exercise.]
3 Pure-Exact Sequences
Pure-Exactness A short exact sequence
0 ! A ˛!B ˇ!C ! 0 (5.3)
is said to be pure-exact if Im ˛ is a pure subgroup of B. Itisp-pure-exact if Im ˛
is p-pure in B.
To simplify notation, in the next theorem we shall use the same letter for
homomorphisms and for maps that they induce.
Theorem 3.1. An exact sequence (5.3) is pure-exact if and only if it satisfies one
.and hence all/ of the following equivalent conditions:
(a) 0 ! nA ˛!nB ˇ!nC ! 0 is exact for every n 2 N;
(b) 0 ! A=nA ˛!B=nB ˇ!C=nC ! 0 is exact for every n;