Abelian Groups - László Fuchs [Springer]

410 12 Torsion-Free Groups
is called the characteristic or the height-sequence of a. We will write A .a/ if
we wish to indicate the group A in which the characteristic was computed. (For
torsion-free groups, transfinite heights are unnecessary: elements of height ! are in
the p-divisible subgroup, so have height 1.)
The following observations are immediate consequences.
(a) . a/ D .a/ for all a 2 A.
(b) If .a/ D .k 1 ; k 2 ;:::;k n ::::)(wherek n 0 or D1), then a 2 A is divisible
by the integer m D p`1
1 p`r r if and only if `i k i for i D 1;:::;r.
(c) If we agree that 1C1 D1,then
.p n a/ D .h p1 .a/; h p2 .a/;:::;h pn .a/ C 1;:::/:
(d) Every sequence .k 1 ; k 2 ;:::;k n ::::/of non-negative integers and symbols 1 is
a realizable characteristic. In the subgroup R of Q that is generated by all pn
`n
with `n k n ,foralln, the element 1 will have this characteristic. With this R,
we can write Ra Dhai if A .a/ D .k 1 ; k 2 ;:::;k n ::::/in a group A.
The set of all characteristics is partially ordered under the pointwise ordering; it
becomes a complete, distributive lattice of the cardinality of the continuum 2 @ 0
.The
lattice operation ^ is the point-wise minimum, and _ is the point-wise maximum.
The minimum member of this lattice is the characteristic .0;0;:::;0;:::/of 1 2 Z,
while the maximum is the characteristic .1; 1;:::;1;:::/of every element in Q.
(e) If b; c 2 A,then.b C c/ .b/ ^ .c/. Thus
A./ Dfa 2 A j .a/ g
is for each characteristic a (fully invariant) subgroup of A. Furthermore,if
A D B ˚ C and b 2 B; c 2 C,then.b C c/ D .b/ ^ .c/.
(f) A subgroup B of a torsion-free group A is pure exactly if B .b/ D A .b/ for all
b 2 B.
(g) If B is a pure subgroup of the torsion-free A, then the characteristic of a coset
mod B is computed according to the rule
A=B .a C B/ D _ b2B
A .a C b/:
(h) If ˛ W A ! C is a homomorphism between torsion-free groups, then A .a/
C .˛a/ for all a 2 A.
For a torsion-free A, the inclusion map Z ! Q induces an embedding A Š
Z ˝ A ! Q ˝ A. The latter group is torsion-free, moreover, a Q-vector space; its
dimension is equal to the rank rk.A/ of A.IfB is a pure subgroup of A,thenA=B is
torsion-free, its rank is called the corank of B in A.