Abelian Groups - László Fuchs [Springer]

586 15 Mixed Groups
We denote by W F ! A the map induced by x a 7! a, andsetH D Ker . Define
T.A/ D G=H, sothatA D F=H is a subgroup in T.A/. IfT.A/ is furnished with
the height valuation, then—as is straightforward to check—the restriction to A will
be identical with v p .Thep-niceness of A in T.A/ follows from that of F in G. The
functorial behavior of the process is evident from the definition. (c) is an obvious
consequence of the definition of Nunke groups.
ut
Global Valuation Turning to the global case, we introduce a valuation where
the values are no longer ordinals and symbols 1, but countable sequences D
. 1 ; 2 ;:::; n ;:::/of ordinals and 1 (like characteristics for torsion-free groups).
Suppose A has p-valuation v p for each prime p.Fora 2 A, weset
.a/ D .v 2 .a/; v 3 .a/;:::;v pn .a/;:::/
where p n stands for the nth prime. The point-wise ordering makes the collection of
these sequences a lattice-ordered class with .0;0;:::;0;:::/as minimum and with
.1; 1;:::;1;:::/ as maximum element. This ‘global valuation’ (which we
might call ‘characteristic’) satisfies conditions (i)–(iv) listed above for p-valuations.
The characteristics (as above) and 0 D .1 0;0 2 ;:::;0 n ;:::/ will be regarded
equivalent, in notation: 0 ,if n D n 0 for almost all n. For instance,
.nx/ .x/ holds for all n 2 N. We will write Πfor the equivalence class of
the characteristic . The collection of these Πcarries a lattice-order.
Let C D hxi be an infinite cyclic group that is furnished with a free global
valuation, given by an arbitrarily chosen characteristic .x/ D . 2 ; 3 ;:::; p ;:::/.
The cyclic groups C Dhxi and C 0 Dhx 0 i are of the same type if the characteristics
.x/ and .x 0 / are equivalent.
Height-Matrices In mixed groups A, we have to consider the overall divisibility;
this can be accomplished by listing the indicators of an element for all primes.
Let p 1 ;:::;p n ;::: denote the sequence of the prime numbers in order of magnitude.
With an element a 2 A we associate the ! ! height-matrix
0
h p1 .a/ h p1 .p 1 a/ ::: h p1 .p k 1 0 1
1a/ ::: u p1
.a/
H.a/ D B ::: ::: ::: ::: :::
C
@ h pn .a/ h pn .p n a/ ::: h pn .p k na/ ::: A D :::
B C
@ u pn
.a/ A Dk nkkI
::: ::: ::: ::: ::: :::
(see Rotman [1], Megibben [3], Myshkin [1]) where the first column is .a/, and
the nth row represents the p n -indicator of a. Keep in mind that the entry nk 2 H.a/
in the .n; k/-position records the p n -height of p k na,foralln 2 N and k