The author determines the isomorphism classes of semisimple Hopf algebras \(A\) of dimension 6, 8 over an algebraically closed field \(k\) such that \((\dim A)1\neq 0\). As conclusions, he shows that such Hopf algebras of dimension 6 consist of 3 isomorphism classes represented by \(kC_6\), \(kS_3\), \(k^{S_3}\) where \(S_3\) denotes the symmetric group of degree 3 and \(C_6\) denotes the cyclic group of order 6; such Hopf algebras of dimension 8 consist of 8 isomorphism classes represented by \(k(C_2\times (C_2\times C_2))\), \(k(C_2\times C_4)\), \(kC_8\), \(kD\), \(k^D\), \(kQ\), \(k^Q\), \(A\), where \(D= C_4\rtimes C_2\) is the dihedral group, \(Q\) is the quaternion group. Among these, \(A\) is the unique one that is neither commutative nor cocommutative and is generated as a Hopf algebra by \(x\), \(y\), \(z\) with some relations. To show this, an important role is played by the biproduct and the bicrossed product.
The classification of semisimple Hopf algebras of various small dimensions is an important subject. The author and some other mathematicians have obtained many results. The author is planning to go on with this work and in particular to complete the classification of all semisimple Hopf algebras of dimensions up to 31.