Community detection is a significant and challenging task in network research. Nowadays, plenty of attention has been focused on local methods of community detection. Among them, community detection with a greedy algorithm typically starts from the identification of local essential nodes called central nodes of the network; communities expand later from these central nodes by optimizing a modularity function. In this paper, we propose a new central node indicator and a new modularity function. Our central node indicator, which we call local centrality indicator (LCI), is as efficient as the well-known global maximal degree indicator and local maximal degree indicator; on certain special network structure, LCI performs even better. On the other hand, our modularity function F2 overcomes certain disadvantages,such as the resolution limit problem,of the modularity functions raised in previous literature. Combined with a greedy algorithm, LCI and F2 enable us to identify the right community structures for both the real world networks and the simulated benchmark network. Evaluation based on the normalized mutual information (NMI) suggests that our community detection method with a greedy algorithm based on LCI and F2 performs superior to many other methods. Therefore, the method we proposed in this paper is potentially noteworthy.