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The Hopf algebra of a uniserial group. (English) Zbl 1067.14041
Let \(k\) be an algebraically closed field of characteristic \(p > 0\). A uniserial group is a finite, commutative, infinitesimal, unipotent \(k\)-group scheme that has a unique composition series. Every uniserial group is either \(F\)-uniserial or \(V\)-uniserial. R. Farnsteiner, G. Röhrle and D. Voigt [Colloq. Math. 89, No. 2, 179–192 (2001; Zbl 0989.16013)] showed that the Dieudonne modules of \(V\)-uniserial groups fall into one of three different types.
The author gives a simple classification of the Dieudonne modules of \(V\)-uniserial groups and connects his classification to that of Farnsteiner et. al. Similar results for \(F\)-uniserial groups follow by duality. The author also determines the representing Hopf algebras. The results are extended to \(k\) a finite field; over finite \(k\) the author determines the number of isomorphism classes of uniserial groups of order \(p^n\) for all \(n\).

MSC:
14L15 Group schemes
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