In this paper, we use elementary linear algebra methods to explore possible Hopf algebra structures within the generalized quaternion algebra. The sufficient and necessary conditions that make the generalized quaternion algebra a Hopf algebra are given. It is proven that not all of the generalized quaternion algebras have Hopf algebraic structures. When the generalized quaternion algebras have Hopf algebraic structures, we describe all the Hopf algebra structures. Finally, we shall prove that all the Hopf algebra structures on the generalized quaternion algebras are isomorphic to Sweedler Hopf algebra, which is consistent with the classification of 4-dimensional Hopf algebras.

In this paper, we will establish a criterion for automorphisms of finite-dimensional algebras. As an application, we will describe all automorphisms of the single-parameter generalized quaternion algebra. Additionally, we will obtain all automorphisms of Sweedler's 4-dimensional Hopf algebra.