Single-index models (SIMs) have been widely used in various applications due to their simplicity and interpretability. However, despite the potential for SIMs to result in discriminatory outcomes based on sensitive attributes like gender, race, or ethnicity, the issue of fairness has not been thoroughly examined in recent studies on the topic. This paper aims to address these fairness concerns by proposing methods for building fair SIMs. Specifically, based on the definition of equal opportunity, we first provide a fairness definition for SIM. Next, we develop a unified fair SIM model and propose an efficient method to solve the fair SIM. Theoretically, we also show that our output is consistent in fairness. Finally, we conduct comprehensive experimental studies over 7 benchmark datasets and demonstrate that our fair SIM outperforms the other 8 baseline methods.