Abelian Groups - László Fuchs [Springer]

76 3 Direct Sums of Cyclic Groups
We can define F formally by starting with a set X Dfx i g i2I of symbols, called
a free set of generators, and declaring F as the set of all formal expressions (3.1)
under the mentioned equality and addition. We say that F is the free group on the
set X.
Example 1.1. An immediate example for a free group is the multiplicative group of positive
rational numbers. The prime numbers form a free set of generators.
Needless to say, F is, up to isomorphism, uniquely determined by the cardinal
number DjIj of the index set I. Thus we are justified to write F for the free group
with free generators. is also called the rank of the free group F, in symbols,
rk F D (for the discussion of rank, see Sect. 4).
Theorem 1.2. The free groups F and F are isomorphic exactly if the cardinals
and are equal.
Proof. We need only verify the ‘only if’ part of the assertion. Observe that if F is
a free group with free generators x i .i 2 I/, thenanelement(3.1) ofF belongs to
pF if and only if pjn 1 ;:::;pjn k . Hence, if p is a prime, then F=pF is a vector space
over the prime field Z=pZ of characteristic p with basis fx i C pFg i2I , and so its
cardinality is p jIj or jIj according as I is finite or infinite. Thus jF=pFj completely
determines jIj.
ut
The Universal Property Free groups enjoy a universal property formulated in
the next theorem which is frequently used for the definition of free groups.
Theorem 1.3 (Universal Property of Free Groups). Let X be a free set of
generators of the free group F. Any function f W X ! A of X into any group A
extends uniquely to a homomorphism W F ! A. This property characterizes free
sets of generators, and hence free groups.
Proof. Write X Dfx i g i2I ,andf .x i / D a i 2 A. There is only one way f can be
extended to a homomorphism W F ! A, namely, by letting
g D .n 1 x i1 CCn k x ik / D n 1 a i1 CCn k a ik :
(The main point is that the uniqueness of (3.1) guarantees that is well defined.) It
is immediate that preserves addition.
To verify the second part, assume that a subset X of a group F has the stated
property. Let G be a free group with a free set Y Dfy i g i2I of generators, where
the index set is the same as for X. By hypothesis, the correspondence f W x i 7! y i
extends to a homomorphism W F ! G; this cannot be anything else than the
map n 1 x i1 CCn k x ik 7! n 1 y i1 CCn k y ik . is injective, because the linear
combination of the y i is 0 only in the trivial case. is obviously surjective, and so it
is an isomorphism.
ut
Mapping X onto a generating system of a given group, we arrive at the following
result which indicates that the group Z is a generator of the category Ab (‘generator’
in the sense used in category theory).