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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
##### Keywords:
Dieudonné module; Witt vectors
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