Hopf algebras which factorize through the Taft algebra T_{m^{2}}(q) and the group Hopf algebra K[C_{n}]

We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra T m 2 (q) and the group Hopf algebra K[C n]: they are nm 2-dimensional quantum groups T ω nm2(q) associated to an n-th root of unity ω. Furthermore, using Dirichlet’s prime number theorem we are able to count the number of isomorphism types of such Hopf algebras. More precisely, if d = gcd(m,v(n)) and {Formula presented} is the prime decomposition of {Formula presented} then the number of types of Hopf algebras that factorize through T m 2 (q) and K[C n] is equal to (α 1 + 1)(α 2 + 1) … (α r + 1), where v (n) is the order of the group of n-th roots of unity in K. As a consequence of our approach, the automorphism groups of these Hopf algebras are described as well.