Abelian Groups - László Fuchs [Springer]

3 Valuated Groups. Height-Matrices 589
do not try hard enough to obtain other kinds of more complicated, but still simply recognizable
invariants. We have to learn more about valuation, and look for invariants in the category V p of
p-valuated p-local groups.
It might, perhaps, be appropriate to mention another kind of valuation used in the literature: the
‘coset valuation.’ Let C be a subgroup of A. Thep-height of the coset a C C in A=C is calculated
ordinarily as sup c2C h p .a C c/ (or 1). In some situations, it is desirable to distinguish according
as this supremum is equal to one of h p .a C c/ (i.e., the coset contains an element proper with
respect to C) or not. This can be done, e.g. by defining the coset valuation as follows: k p .a C C/ D
sup c2C .h p .a C c/ C 1/.
Grinshpon–Krylov [1] published an extensive study of transitivity and full transitivity in mixed
groups. A is fully transitive if for all x; y 2 A with H.x/ H.y/, thereisan 2 End A such that
.x/ D y. Inter alia, the full transitivity of direct sums is investigated. See also Misyakov [1] and
Grinshpon–Misyakov [1].
Let us mention here some literature on universal embeddings. A group U is (purely) universal
for a set S of groups, if U 2 S and every group in S embeds as a (pure) subgroup in U. Kojman–
Shelah [1] consider various classes of interest; their main set-theoretic tools are club guessing
sequences.
Exercises
(1) Let A be p-local, and v p a p-valuation in A. For 2 , defineA./ Dfa 2 A j
v p .a/ g. Show that the collection of the A./ determine v p .
(2) Let A be a p-local mixed group. The indicators of elements in A satisfy the gap
condition (see Sect. 1 in Chapter 10).
(3) Prove the inequality for height-matrices: H.a C b/ H.a/ ^ H.b/ for all
a; b 2 A.
(4) Define A Dha 0 ; a 1 ;:::;a n ;:::i subject to the relations p n a 0 D p k n
a n for n 1,
where 0