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We prove that a Hopf algebra of prime dimension   p   over an algebraically closed field, whose characteristic is equal to   p  , is either a group algebra or a restricted universal enveloping algebra. Moreover, we show that any Hopf algebra of prime dimension   p   over a field of characteristic    q  >  0    is commutative and cocommutative when    q  =  2    or    p  <  4  q   . This problem remains open in positive characteristic when    2  <  q  <  p  /  4   .

Hopf algebras of prime dimension in positive characteristic