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\(p\)-adic order bounded group valuations on Abelian groups. (English) Zbl 1133.16028
For a finite group \(G\), and a local field \(K\) with ring of integers \(R\) and uniformizer \(\pi\), a \(p\)-adic order bounded group valuation is a function \(\xi\colon G\to\mathbb{Z}\cup\{\infty\}\) which takes its finite values on \(G\setminus\{1\}\) and satisfies \(\xi(gh)\geq\min\{\xi(g),\xi(h)\}\) like ordinary valuations, as well as \[ \xi([g,h])\geq\xi(g)+\xi(h),\quad\xi(g^p)\geq p\xi(g), \] together with an inequality depending on the arithmetic of \(K\). To any such valuation, the \(R\)-algebra generated by the elements \(\pi^{-\xi(g)}(1-g)\) is a Hopf order in \(KG\) [see R. G. Larson, J. Algebra 38, 414-452 (1976; Zbl 0407.20007)]. For \(G\) Abelian, the authors construct the \(p\)-adic order bounded group valuations \(\xi\). They characterize \(\xi\) in terms of the corresponding chain of subgroups of \(G\). Calculations are provided for cyclic or elementary Abelian groups \(G\). An explicit description of the associated Larson orders is given for \(p\)-groups \(G\) of order \(\leq p^2\).

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16S34 Group rings
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