Let 𝔉 be a class of finite groups. A subgroup H of G is said to be 𝔉-supplemented in G if there exists a subgroup T of G such that G = TH and (H ∩ T)HG/HG is contained in the 𝔉-hypercenter [Formula: see text] of G/HG. In this paper, we investigate further the influence of 𝔉-supplemented subgroup on the structure of finite groups and generalize a series of known results.

With the rise of social network platforms such as WeChat, Weibo, and TikTok, social networks have developed from simple social networks to complex social networks. Researchers have gradually found that the traditional data sampling methods can no longer meet the development needs of the complex social network structure. In order to save network resources, various methods of social relationship prediction have been proposed. In this paper, we propose a BTCS algorithm based on low sampling rate under cognitive model and conduct several sets of comparison experiments under different networks and different sampling rates, and the results show that the BTCS algorithm improves the prediction accuracy and reduces the prediction time under low sampling rate. To address the problems of poor stability and slow prediction speed of random sampling prediction methods, this paper proposes a CCS algorithm in colleges and universities using the characteristics of high awareness among nodes within the same college. It can effectively combine the cognitive characteristics of the nodes with the college attributes and apply them to the relationship prediction to realize the college-oriented relationship prediction. The simulation results show that the CCS algorithm is more stable than other random sampling prediction methods. The results make full use of the cognitive characteristics and college attributes of nodes in social networks; reduce the influence of multiple factors such as response time, data packet loss, and individual behavior on relationship prediction; and improve the efficiency of college student group relationship prediction, which has certain theoretical significance and application prospects.