Abelian Groups - László Fuchs [Springer]

4 Linear Independence and Rank 93
to this property, then the cardinality of the system is called the torsion-free rank
rk 0 .A/ (p-rank rk p .A/)ofA. From the definitions it is evident that the equation
rk.A/ D rk 0 .A/ C X p
rk p .A/ (3.12)
holds with p running over all primes. Obviously, rk.A/ D 0 means A D 0.
At this point the natural question is: how unique are these various ranks? In order
to legitimize them, we need to show:
Theorem 4.3. The ranks rk.A/; rk 0 .A/; rk p .A/ of a group A are invariants of A.
Proof. It suffices to prove that rk 0 .A/ and rk p .A/ are independent of the choice of
the maximal independent system defining them.
It is routine to check that rk 0 .A/ D rk.A=tA/. As a consequence, in proving the
invariance of rk 0 .A/, we may assume without loss of generality that A is a torsionfree
group. Let fa 1 ;:::;a k g and fb 1 ;:::;b`g be two maximal independent systems
in A. Then there are integers m; m i ; n; n j with m; n ¤ 0 such that ma i D P`
jD1 m ijb j
and nb j D P k
iD1 n jia i . Hence
mna i D
kX
hD1 jD1
`X
n ij m jh a h
where the corresponding coefficients on both sides must be equal. This means that
the product of matrices kn ij kkm jh k is a scalar matrix mnE k (E k denotes the k k
identity matrix). This is impossible if k