The mobile edge computing has emerged as a promising architecture, which reduces computation delay and communication cost to optimize the quality of services for users. Biometric-based authentication is an important security technology to secure edge computing user services. However, in existing authentication studies, the user only binds biometrics with one authorized device. Authorizing multiple mobile terminals to users requires additional registration phases. The system security and reliability will inevitably be influenced. To address this issue, we propose privacy-preserving blockchain-based user authentication with device group authorization. Specifically, the blockchain technology is leveraged to authorize device group to users and support distributed edge computing environments. In addition, the combination of group signature and biometrics simultaneously authenticates the user and the holding authorized terminal. The security discussions show the security features and functionalities. We also implemented a prototype to perform experimental evaluation. The results prove the efficiency and practicality of our work.