Abelian Groups - László Fuchs [Springer]

1 Purity 151
called the pure subgroup generated by S. It is easy to check that hSi =hSi is
precisely the torsion subgroup in A=hSi.
(H) Purity is an inductive property: the union of a chain of pure subgroups is pure.
For, if G is the union of a chain G 1 G ::: of pure subgroups, and
if nx D g 2 G is solvable in A, then it is solvable in G for every index with
g 2 G .Itisa fortiori solvable in G.
The following theorem is of utmost importance, it lists the most frequently
needed properties of purity.
Theorem 1.3 (Prüfer [2]). Let B; C be subgroups of the group A such that C
B A. We then have:
(i) if C is pure in A, then it is pure in B;
(ii) if C is pure in B and B is pure in A, then C is pure in A;
(iii) if B is pure in A, then B=C is pure in A=C;
(iv) if C is pure in A and B=C is pure in A=C, then B is pure in A.
Proof.
(i) is obvious.
(ii) Under the stated hypotheses, nC D C \ nB D C \ .B \ nA/ D .C \ B/ \ nA D
C \ nA for every n 2 N.
(iii) follows from the equalities n.B=C/ D .nB C C/=C D Œ.B \ nA/ C C=C D
ŒB \ .nA C C/=C D B=C \ n.A=C/ (we used the modular law).
(iv) Assuming the stated hypotheses, let na D b 2 B for some a 2 A and n 2 N.
Then n.a C C/ D b C C whence hypothesis implies that there is a b 0 2 B such
that n.b 0 C C/ D b C C, i.e.nb 0 D b C c for a suitable c 2 C. Inviewofthe
purity of C, from n.b 0 a/ D c we get a c 0 2 C satisfying nc 0 D c. It only
remains to check that b 0 c 0 2 B and n.b 0 c 0 / D b. ut
Thus (iii) and (iv) combined claim that the natural correspondence between
subgroups of A=C and subgroups of A containing the pure subgroup C preserves
purity.
Lemma 1.4. Let B be a pure subgroup of A. If B is torsion-free and A=B is torsion,
then B is a summand of A.
Proof. If T denotes the torsion subgroup of A, thenB ˚ T is an essential subgroup
in A. We claim that it is all of A. Foreverya 2 A n T there is an integer n >0such
that na D b C t with b 2 B; t 2 T. Forsomem >0we have mt D 0,somna D mb.
By purity, some b 0 2 B satisfies mna D mnb 0 .Thena b 0 2 T and a 2 B C T. ut
Embedding in Pure Subgroup A fundamental property of purity is stated in
the following theorem.
Theorem 1.5 (Szele). Every finite subgroup can be embedded in a countable pure
subgroup, and every subgroup of infinite cardinality in a pure subgroup of the same
power.