Abelian Groups - László Fuchs [Springer]

156 5 Purity and Basic Subgroups
Proof. If the group contains a quasi-cyclic group Z.p 1 / for some prime p, then this
is a direct summand. If it does not contain any quasi-cyclic subgroup, but it contains
elements of order p, then it must also contain one of finite height; cf. (E). The claim
follows from Corollary 2.2.
ut
Another corollary worthwhile recording is as follows.
Corollary 2.4 (Kulikov [1]). A directly indecomposable group is either cocyclic
or torsion-free.
Proof. By Corollary 2.3 there are no indecomposable mixed groups, and the only
indecomposable torsion groups are the cocyclic groups.
ut
Bounded Pure Subgroups We now proceed to show that certain pure subgroups
are necessarily summands.
Theorem 2.5 (Prüfer [2], Kulikov [1]). A bounded pure subgroup is a direct
summand.
Proof. If B is a bounded group, then Lemma 2.1 allows us to decompose B D B 1˚C
where B 1 is a direct sum of cyclic groups of the same order p k , and the bound for C
is less than the bound for B. IfB is pure in A, thensoisB 1 , and Lemma 2.1 implies
that A D B 1 ˚ A 1 for some A 1 A. ThenB D B 1 ˚ C 1 with C 1 D B \ A 1 Š C.
Here C 1 is pure in A 1 , and by induction, C 1 is a summand of A 1 , and hence B is
one of A.
ut
Corollary 2.6 (Khabbaz [2]). p n A-high subgroups .n 2 N/ of a group A are
summands.
Proof. We show that a p n A-high subgroup B is a bounded pure subgroup. Since
p n B B\p n A D 0, B is a bounded p-group. By induction, we prove that B\p k A
p k B for integers k with 0 k n. This being trivially true for k D 0, assume it is
true for some k where 0 k < n. Letb D p kC1 a ¤ 0.b 2 B; a 2 A/. Ifp k a 2 B,
then by induction hypothesis p k a D p k b 0 for some b 0 2 B, andthenb D p kC1 b 0 .If
p k a … B, thenhB; p k ai contains a non-zero element b 0 C p k a 2 p n A,whereb 0 2 B
is also in p k A,sob 0 2 p k B as well. Now b 0 D p n a 0 p k a for an a 0 2 A, and
hence b D p kC1 a D p nC1 a 0 pb 0 ; where the term p nC1 a 0 has to vanish because of
B \ p n A D 0. Thus b D pb 0 2 p kC1 B: ut
Corollary 2.7 (Erdélyi [1]). A p-subgroup embeds in a bounded summand if and
only if the heights of its elements .computed in the containing group/ are bounded.
Proof. The necessity being obvious, suppose that p m is an upper bound for the
heights in the subgroup B of A.Thereisap mC1 A-high subgroup C such that B C.
Invoke the preceding corollary to conclude that C is a summand of A. ut
Cosets of Pure Subgroups Here is another characterization of purity.
Lemma 2.8 (Prüfer [1]). A subgroup B of A is pure if and only if every coset of A
modulo B can be represented by an element that has the same order as the coset.