Abelian Groups - László Fuchs [Springer]

3 Torsion-Complete Groups 317
Exercises
(1) The torsion part of a direct product of p-groups is torsion-complete if and
only if each summand is torsion-complete.
(2) Using the notation of the text, prove that . Q n B n/=B is a torsion-free
algebraically compact group which is not divisible unless 0.
(3) Large subgroups in torsion-complete groups are likewise torsion-complete.
(4) Using the notation in the text, an element g D .b 1 ;:::;b n ;:::/ 2 B of order
p n generates a cyclic summand of B if and only if o.b n / D p n .
(5) (a) Let A be a separable p-group, and B an upper basic subgroup of A. If
B ¤ A,thenjA=Bj @ 1 , and there is a decomposition A D A 0 ˚ A 00 such
that A 0 is †-cyclic and jA 00 jDjA=Bj. [Hint: argue as in Theorem 3.5.]
(b) Every separable p-group A can be written as A D A 0 ˚ A 00 such that A 0 is
†-cyclic, and every basic subgroup of A 00 is both upper and lower.
(6) In a torsion-complete p-group, all basic subgroups are lower as well as upper.
[Hint: Theorem 3.11.]
(7) (Kemoklidze) If B is a basic subgroup of the p-group A, and if every
endomorphism of B extends to A, then either A D B or A D B.
(8) Let B be a torsion-complete p-group with fin rk B D . There is a direct
decomposition B D G ˚ H, whereH is bounded and G is torsion-complete
of cardinality @ 0
.
(9) Kernels (but not all images) of endomorphisms of torsion-complete groups
are again torsion-complete.
(10) (Faltings) A p-group A is isomorphic to the torsion part of Hom.A; Z.p 1 //
if and only if it is torsion-complete with finite UK-invariants.
(11) Let A be a separable p-group with basic subgroup B.
(a) If B=A Š .Z.p 1 // ./ ,thenPext.Z.p 1 /; A/ Š A .Jp / ./ .
(b) The Ulm factors of Ext.Z.p 1 /; A/ are: QB and
A .Jp / ./ .
(12) (Richman) Let A be a separable p-group, and G a bounded subgroup. A is
torsion-complete if and only if so is A=G. [Hint: Corollary 3.8.]
(13) Every height-preserving automorphism of the socle of B can be extended to
an automorphism of B.
(14) (Megibben) Let A be an unbounded torsion-complete p-group, and C an
arbitrary separable p-group. There exists a homomorphism of A into C which
is not small exactly if C contains an unbounded torsion-complete subgroup.
[Hint: exhibit an unbounded torsion-complete group in the image of a nonsmall
homomorphism.]
(15) Let T be a pure dense subgroup in the torsion-complete p-group B containing
the basic subgroup B such that rk.B=T/ D n 2 N. Then
(a) Pext.Z.p 1 /; T/ Š˚nJ p ;and
(b) if S is a separable p-group containing T as a pure dense subgroup with
rk.S=T/ D n,thenS Š B.