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\(n\)-summable valuated \(p^n\)-socles and primary Abelian groups. (English) Zbl 1210.20049

Let \(V\) be a \(p\)-primary valuated Abelian group. The induced valuation on the subgroup \(V[p^n]\) has the following property (*) If \(x\in V[p^{n-1}]\) and \(\beta\) is an ordinal (strictly) less than the value \(v(x)\), then there is \(y\in V[p^n]\) such that \(py=x\) and \(v(y)\geq\beta\).
Motivated by this observation the authors introduce ‘valuated \(p^n\)-socles’ as being \(p^n\)-bounded valuated groups \(V=V[p^n]\) satisfying the condition (*). They develop a theory of these socles that deals with “\(n\)-versions” of concepts from \(p\)-group theory such as summable, totally projective, Ulm invariant, balanced, isotype, nice, and some variants of these.
Sample results are as follows.
Lemma 1.11. Let \(V\) be a valuated \(p^n\)-socle and \(W\) an \(n\)-balanced subgroup of \(V\). If \(V/W\) is countable, then there is a valuated decomposition \(V=W\oplus C\) where \(C\) is a countable \(n\)-isotype subgroup of \(V\).
Theorem 2.1. Let \(V\) be a valuated \(p^n\)-socle of length \(\leq\omega_1\). Then \(V\) is \(n\)-summable if and only if \(V\) has a nice system if and only if \(V\) has a nice composition series.
Corollary 2.3. Let \(V\) be a valuated \(p^n\)-socle and \(W\) an \(n\)-isotype subgroup of \(V\). If \(W\) is \(n\)-summable and \(V/W\) is countable, then \(V\) is \(n\)-summable.
Proposition 2.6 and Theorem 2.7. Suppose that the valuated \(p^n\)-socles \(V\) and \(W\) are either countable or \(n\)-summable. Then \(V\) and \(W\) are isometric if and only if they have the same Ulm invariants.
Applications to \(p\)-groups include the following.
Theorem 3.8. A reduced group \(G\) is \(n\)-summable for every \(1\leq n<\omega\) if and only if it is a direct sum of countable groups.
Corollary 3.9. A reduced group \(G\) is strongly \(n\)-summable for every \(1\leq n<\omega\) if and only if it is a direct sum of cyclic groups.

MSC:

20K10 Torsion groups, primary groups and generalized primary groups
20K25 Direct sums, direct products, etc. for abelian groups
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References:

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