Abelian Groups - László Fuchs [Springer]

374 11 p-Groups with Elements of Infinite Height
ρφ :
: 0 −−−−−→ B
μ
0 −−−−−→ B/p ρ B
∥
ρ : 0 −−−−−→ B/p ρ B
α
−−−−−→
γ
−−−−−→
ν
G
H
⏐
ψ↓
ɛ
−−−−−→ G/p ρ G
β
−−−−−→ A −−−−−→ 0
∥
δ
−−−−−→ A −−−−−→ 0
⏐
↓φ
ξ
−−−−−→ A/p ρ A −−−−−→ 0,
We get the middle row as a pull-back. The canonical map W G ! G=p G
followed by equals ˇ, so the pull-back property ensures the existence of a
unique W G ! H such that D and ı D ˇ. If W B ! B=p B denotes
the canonical map, then for the commutativity of the diagram it only remains to
check that D ˛.Now D D ˛ D ˛ and ı D 0 D ˇ˛ D ı˛
show that and ˛ are maps B ! H which become equal if followed by or ı.
Since H was a pull-back, D ˛ follows.
By a remark in Sect. 5, after (11.7), B=p B is balanced in G=p G, and
therefore e must split (by induction hypothesis, A=p A is balanced-projective).
It follows that e also splits, and hence in view of the exact sequence
Ext.A; p
B/ !Ext.A; B/
!Ext.A; B=p B/ we conclude that e 2 Im where
W p B ! B is the inclusion map. By Proposition 6.2, Im p Ext.A; B/, and
therefore e 2\ < p Ext.A; B/ D p Ext.A; B/. The last group is 0, if A is totally
projective, i.e. e is splitting.
ut
Before continuing, let us summarize briefly the various characterizations of the
class of p-groups we are studying. The theorems we have proved so far already show
that for a reduced p-group A the following are equivalent:
1. A is simply presented;
2. A has a nice system (or a nice composition chain);
3. A is a summand of a direct sum of generalized Prüfer groups;
4. A is balanced-projective;
5. A is totally projective.
And more importantly, these groups are characterized by their UK-invariants.
Though these groups can be referred to by any of the listed properties, we
more often use the term ‘totally projective p-groups,’ because chronologically
this property was first introduced. However, total projectivity is perhaps the least
practical criterion to recognize groups in this class.
Consequences of the Main Result Capitalizing on Theorem 6.5, we can get
valuable information about how totally projective p-groups are related to direct sums
of countable groups.
Theorem 6.6 (Nunke [5]). A reduced p-group is a direct sum of countable groups
if and only if it is totally projective of length ! 1 .