Abelian Groups - László Fuchs [Springer]

1 Freeness and Projectivity 77
Corollary 1.4. Every group with at most generators is an epimorphic image of a
free group with generators.
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Consequently, every group A can be embedded in a short exact sequence
0 ! H ! F !A ! 0;
where F is free group, and H D Ker . (We will see shortly that H is likewise free.)
This is called a free resolution of A. It is far from being unique, because both F and
can be chosen in many ways.
If is an infinite cardinal, then F has 2 subsets, and hence at most 2 subgroups
and factor groups. We conclude that there exist at most 2 pairwise non-isomorphic
groups of cardinality . (We will learn in Corollary 3.8 in Chapter 11, that2 is
the precise number.)
The next two theorems are fundamental, they are quoted most frequently.
Theorem 1.5. Suppose that B is a subgroup of a group A such that A=B isafree
group. Then B is a summand of A, i.e., A D B ˚ C for a subgroup C Š A=B.
Proof. That only free factor groups can share the stated property will follow
from Theorem 1.7. In order to show that free groups do have this property,
by Lemma 2.4 in Chapter 2, it suffices to verify the claim for A=B Š Z only, say
A=B D ha C Bi with a 2 A. The elements of A=B are the cosets n.a C B/ D
na C B .n 2 Z/ (all different). Hence A D B ˚hai is immediate.
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This theorem can also be phrased by saying that an exact sequence 0 ! B !
A ! F ! 0 with a free group F is necessarily splitting.
Subgroups of Free Groups In the next theorem we study the subgroups of free
abelian groups. Recall the famous result in group theory that subgroups of (noncommutative)
free groups are again free. For abelian groups the situation is the
same. To prove this, we use a well ordering of the index set.
Theorem 1.6. Subgroups of free groups are free.
Proof. Let F be a free group on the set X, which we now assume to be well ordered,
say X Dfx g < for some ordinal . Thus F D˚< hx i.For < ,define
F D˚< hx i,andsetG D G \ F for a subgroup G < F. Clearly, G D
G C1 \ F ,soG C1 =G Š .G C1 C F /=F . The last factor group is a subgroup
of F C1 =F Šhx i; thus either G C1 D G or G C1 =G is an infinite cyclic group.
From Theorem 1.5 we conclude that G C1 D G ˚hg i for some g 2 G C1
(which is 0 if G C1 D G ). It follows that the elements g generate the direct sum
˚< hg i in G. This must be all of G, sinceG is the union of the G . < /. ut