LAUSR

. We extend [DGNO1, Theorem 4.5] and [LKW, Theorem 4.22] to positive characteristic (i.e., to the finite, not necessarily fusion, case). Namely, we prove that if D is a finite nondegenerate braided tensor category over an algebraically closed field k of characteristic p > 0, containing a Tannakian Lagrangian subcategory Rep( G ), where G is a finite k -group scheme, then D is braided tensor equivalent to Rep( D ω ( G )) for some ω ∈ H 3 ( G, G m ), where D ω ( G ) denotes the twisted double of G [G]. We then prove that the group M ext (Rep( G )) of minimal extensions of Rep( G ) is isomorphic to the group H 3 ( G, G m ). In particular, we use [EG2, FP] to show that M ext (Rep( µ p )) = 1, M ext (Rep( α p )) is infinite, and if O (Γ) ∗ = u ( g ) for a semisimple restricted p -Lie algebra g , then M ext (Rep(Γ)) = 1 and M ext (Rep(Γ × α p )) ∼ = g ∗ (1) .