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Knice subgroups of mixed groups. (English) Zbl 0653.20057

Abelian group theory, Proc. 3rd Conf., Oberwolfach/FRG 1985, 89-109 (1987).
[For the entire collection see Zbl 0638.00013.]
Let G be an abelian group, \({\mathbb{Z}}\) the ring of rational integers, \({\mathbb{P}}\) the set of primes in \({\mathbb{Z}}\), \(O_{\infty}\) the class of ordinals with the symbol \(\infty\) adjoined, \(| x|\) the sequence of p-heights \((| x|_ p)_{p\in {\mathbb{P}}}\), \(\| x\|\) the height matrix of x in G, that is, \(\| x\|\) is the doubly infinite matrix indexed by \(P\times \omega\) and having \(| p\) ix\(| p\) as its (p,i) entry. More generally a height matrix is any doubly infinite matrix M indexed by \(P\times \omega\) having as its (p,i) entry an element \(m_{p,i}\) of \(O_{\infty}\) and satisfying the condition \(m_{p,i}<m_{p,i+1}\) for all (p,i), similarly a height sequence is \({\bar \alpha}=(\alpha_ i)_{i<\omega}\) where \(\alpha_ i\in O_{\infty}\) and \(\alpha_ i<\alpha_{i+1}\) for all \(i<\omega\), \(M_ p\) is the pth row of M. Letting tG denote the maximal torsion subgroup of G and \({\bar \infty}\) the height matrix having \(\infty\) for each of its entries.
Let \(G(M)=\{x\in G|\) \(\| x\| \geq M\}\); for \({\bar \alpha}=(\alpha_ i)_{i<\omega}\) G(\({\bar \alpha}\),p)\(=\{x\in G|\) \(| p\) \(ix|_ p\geq \alpha_ i\) for all \(i<\omega \}\); G(M \(*)=\{<x\in G(M)|\) \(\| x\| \nsim M>\) if \(M\nsim {\bar \infty}\), G(M \(*)=\{tG\cap G(M)\) if \(M\sim {\bar \infty}\); G(\({\bar \alpha}\) \(*,p)=<x\in G({\bar \alpha},p)|\) \(| p\) \(ix|_ p\neq \alpha_ i\) for infinitely many values of \(i>\); G(M \(*,p)=G(M)\cap (G(M\) \(*)+G(M\) \(*_ p,p)).\)
A subgroup \(H\subseteq G\) is said to be nice if \((p^{\alpha}(G/H)/(p^{\alpha}G+H)/H)[p]=0\) for all ordinals \(\alpha\) and all primes \(p\in {\mathbb{P}}\). An element x of G is called primitive if for each positive integer n, each height matrix M and each prime p nx\(\in G(M\) *,p) implies \(\| x\| \nsim M\) or \(| p\) \(inx|_ p\neq m_{p,i}\) for infinitely many values of i. Let \(A=\oplus_{i\in I}A_ i\) where \(\{A_ i\}_{i\in I}\) is a family of independent subgroups of G. For each \(a\in A\), write \(a=\sum_{i\in I}a_ i\) where \(a_ i\in A_ i\) for all i. The group A is said to be a *-valuated coproduct in G provided, for each \(a\in A\) and for all height matrices M, height sequences \({\bar \alpha}\) and primes p, the following conditions are satisfied:
(1) If \(a\in G(M)\), then \(a_ i\in G(M)\) for all i;
(2) If \(a\in G(M\) *), than \(a_ i\in G(M\) *) for all i;
(3) If \(a\in G({\bar \alpha}\) *,p), then \(a_ i\in G({\bar \alpha}\) *,p) for all i;
(4) If \(a\in G(M\) *,p), then \(a_ i\in G(M\) *,p) for all i.
If only condition (1) is assumed, then A is said to be valuated coproduct in G.
A subgroup A of G is called knice provided A is nice in G and if S is a finite subset of G, then there is a finite collection of primitive elements \(y_ 1,...,y_ m\) and a positive integer n such that \(A\oplus <y_ 1>\oplus...\oplus <y_ m>\) is a *-valuated coproduct that contains \(n<S>.\)
The main results of this paper are: Theorem 1.5. A finite extension of a nice subgroup is again nice. - Theorem 1.7. If B is nice in G and if \(H=B\oplus <c>\) is a valuated coproduct in G, then H is a nice subgroup of G. Theorem 2.10. Let both \(A\oplus <y_ 1>\) and \(B=A\oplus <x_ 1>\oplus...\oplus <x_ n>\) be *-valuated coproducts in G where \(y_ 1\) and all the \(x_ i's\) are primitive. If \(y_ 1\in B\), then there are primitive elements \(y_ 2,...,y_ n\) such that \(B'=A\oplus <y_ 1>\oplus...\oplus <y_ n>\) is a *-valuated coproduct with B/B’ finite. - Theorem 3.2. If A is a knice subgroup of G and if A’ is a finite extension of A in G, then A’ is also knice in G. - Theorem 3.3. If \(A'=A\oplus <x_ 1>\oplus...\oplus <x_ n>\) is a *-valuated coproduct with each \(x_ i\) primitive in G and if A is a knice subgroup of G, then A’ is also a knice subgroup of G. - Theorem 3.7. If A and B/A are knice subgroups of G and G/A respectively, then B is knice in G.
Reviewer: A.M.Sebel’din

MSC:

20K27 Subgroups of abelian groups
20K21 Mixed groups
20K25 Direct sums, direct products, etc. for abelian groups

Citations:

Zbl 0638.00013