Abelian Groups - László Fuchs [Springer]

316 10 Torsion Groups
Theorem 3.11 (Leptin [1], Enochs [3]). A reduced p-group A is torsion-complete
if and only if every isomorphism between basic subgroups extends to an automorphism
of A.
Proof. In order to verify the ‘if’ part, suppose A has the stated property. If A is
bounded, then it has only one basic subgroup, viz. itself, and there is nothing to
prove. So assume A is unbounded, and let B be a basic subgroup of A. AsB is
unbounded, B < B. LetB 0 be a subgroup of B such that B < B 0 < B and B 0 =B
is countable divisible. Because of Theorem 3.5, B 0 is again a basic subgroup of B,
so by Theorem 3.10 there is an automorphism ˇ of B such that ˇB 0 D B. Then
ˇB.< B/ is also basic in B,andinA as well, so by hypothesis, there is an ˛ 2 Aut A
such that ˛ B D ˇ B.
Defineamap W B ! A as follows. Put b D b for every b 2 B.Ifx 2 BnB,then
it is included in some B 0 ,andwesetx D ˛ 1ˇx. This definition is unambiguous:
if B 00 is another basic subgroup of B containing B 0 ,andifˇ0 2 Aut B with ˇ0B 00 D B
and ˛0 2 Aut A with ˛0 B D ˇ0 B,thenˇ0ˇ 1B D ˇ0B 0 ˇ0B 00 D B and ˛0˛ 1
agrees with ˇ0ˇ 1 on ˇB D ˛B, and hence on B. We conclude that ˛0˛ 1 D ˇ0ˇ 1,
showing that ˛ 1ˇ D ˛0 1ˇ0,sothat is well defined. If x 2 Ker ,then˛ 1ˇx D
0, ˇx D 0; and x D 0,thatis, is monic.
Therefore, B is a subgroup of A such that B D B. This means that B is pure
in A, andsoTheorem3.6(iv) implies that it is a direct summand of A. ButA is
reduced and A=B is divisible, thus necessarily B D A. This proves that is an
isomorphism, and A is torsion-complete.
ut
F Notes. Only a few classes of separable p-groups are well explored. In the countable case,
by Prüfer’s theorem, they are just †-cyclic groups, so the focus should be on the uncountable case.
No general structure theorem is available, and none is expected, but much is known about a few
classes with special properties (torsion-complete, quasi-complete, summable, etc. p-groups).
The theory of torsion-complete p-groups was developed by Kulikov [2], proving many of the
essential results. He also proved that any two direct decompositions of a torsion-complete group
have isomorphic refinements. Observe that the cardinality of any torsion-complete p-group A is
‘essentially’ of the form @0 for some . What we mean is that if A D G ˚ H with bounded H,
and DjGj is minimal, then D @0 .
There is a large body of work on torsion-complete groups, and numerous characterizations
exist for torsion-completeness. Waller [1] considers generalized torsion-completeness to p-groups
of countable limit length . Torsion-completion in the p -topology retains some of the pleasant
features of torsion-complete groups.
In retrospect, it is no surprise that torsion-completeness attracted so much attention, even before
the theory of algebraically compact groups was developed. The similarity of the two theories is
understandable, but, as far as importance is concerned, algebraic compactness is overwhelming.
It is instructive to study generalizations of separability involving higher cardinals. A p-group is
called -separable, if every subgroup of cardinality