Weakly-Supervised Group Activity Recognition (WSGAR) aims to understand the activity performed together by a group of individuals with the video-level label and without actor-level labels. We propose Flow-Assisted Motion Learning Network (Flaming-Net) for WSGAR, which consists of the motion-aware actor encoder to extract actor features and the two-pathways relation module to infer the interaction among actors and their activity. Flaming-Net leverages an additional optical flow modality in the training stage to enhance its motion awareness when finding locally active actors. The first pathway of the relation module, the actor-centric path, initially captures the temporal dynamics of individual actors and then constructs inter-actor relationships. In parallel, the group-centric path starts by building spatial connections between actors within the same timeframe and then captures simultaneous spatio-temporal dynamics among them. We demonstrate that Flaming-Net achieves new state-of-the-art WSGAR results on two benchmarks, including a 2.8%p higher MPCA score on the NBA dataset. Importantly, we use the optical flow modality only for training and not for inference.