Abelian Groups - László Fuchs [Springer]

4 Pure-Injective Hulls 199
(2) The group R of reals admits infinitely many different linearly compact
topologies.
(3) In an algebraically compact group, a finite set of elements is contained in a
summand that is a finite direct sum of Q, cocyclic groups, and copies of J p for
various primes p.
(4) Determine the invariants of the algebraically compact group ΠQ 1
nD1 Z.pn /
where is an infinite cardinal.
(5) (Leptin) A group with a linear topology is linearly compact if and only if it
is complete in this topology, and the factor groups modulo open subgroups
satisfy the minimum condition. [Hint: (e).]
(6) If A; B are algebraically compact groups such that A˚A Š B˚B,thenA Š B.
(7) If A and B are algebraically compact groups, each isomorphic to a pure
subgroup of the other, then A Š B. [Hint: Theorem 3.2.]
(8) J p has the cancellation property: if A; B are groups such that A ˚ J p Š B ˚ J p ,
then A Š B. [Hint: reduce the proof to reduced torsion-free groups.]
(9) (Balcerzyk) Prove that Z N =Z .N/ Š D ˚ Qp A p,whereD is a Q-vector space
of dimension 2 @ 0
,andA p is the p-adic completion of ˚2@ 0 J p . [Hint: the group
Z N =.pZ N C Z .N/ / has cardinality 2 @ 0
.]
(10) Are quasi-injective groups algebraically compact?
(11) The direct sum ˚@0 Z.p n / for any fixed n 2 N is algebraically compact, but it
cannot be linearly compact under any linear topology.
4 Pure-Injective Hulls
The striking analogy between injective and pure-injective groups can be pushed
further by pointing out the analogue of the injective hull.
Pure-Essential Extensions We first introduce the following notation. For a pure
subgroup G of A, K.G; A/ will denote the set of all subgroups H A such that
(i) G \ H D 0;and
(ii) .G C H/=H is pure in A=H.
Since (i) implies G C H D G ˚ H, condition (ii) amounts to the fact that if
nx D g C h .n 2 N/ with g 2 G; h 2 H is solvable for x in A, theng is divisible by
n in G. Hence the set K.G; A/ is closed under taking subgroups. The purity of G in
A assures that 0 2 K.G; A/,soK.G; A/ is never empty.
Moreover, the set K.G; A/ is inductive. To prove this, let H i .i 2 I/ be a chain of
subgroups in K.G; A/, andH their union. H obviously satisfies (i). Suppose nx D
g C h .g 2 G; h 2 H/ is solvable in A. Forsomei 2 I, h 2 H i 2 K.G; A/ whence
g 2 nG,andsoH 2 K.G; A/.
Following Maranda [1], we call a group A a pure-essential extension of its pure
subgroup G, andG a pure-essential subgroup of A, ifK.G; A/ D f0g. A is a
maximal pure-essential extension of G if it is a pure-essential extension, but no