The main result of the paper asserts that in positive characteristic the quantum group \(SL_q(2)\) is geometrically reductive for any \(q\). To prove this the authors extent the concept of geometrically coreductive Hopf algebras to not necessarily commutative Hopf algebras and establish various characterizations of such Hopf algebras. The authors also show that the Hopf algebra \(K[SL_{-1}(2)]\) is geometrically coreductive in any characteristic.