Energy-efficient transmission is an important issue for wireless communication systems, due to the increased energy consumption and limited battery capacity. In this paper, we study the energy-efficient transmission in a single cellular downlink multi-user Orthogonal Frequency Division Multiple Access (OFDMA) system. The transmit power consumption and the circuit power consumption are both taken into consideration, when optimizing the total bits transmitted per Joule of energy. This optimization problem has nonlinear constraints, which are commonly solved by Lagrange-based algorithm. However, these methods are time-consuming and are not optimal. To address this issue, we propose an adaptive subcarrier, bit, and power allocation scheme to optimize energy-efficient transmissions based on genetic algorithms. Moreover, we improve the genetic algorithm with a new elitist method and a penalty handling method that are specific to our optimization problem. The simulation results show that the proposed scheme can obtain more satisfied solutions in a shorter time, compared to the traditional methods.