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The two-receiver Gaussian broadcast channel is a model of a communication system where a single codeword is transmitted over two distinct Gaussian channels that have different signal-to, noise ratios. The receiver with the better signal-to-noise ratio decodes all of the information carried by the codeword, while the other receiver decodes only some of the information. The concept of group codes for the Gaussian broadcast channel is developed and applied to permutation codes for the Gaussian broadcast channel. A group code for the Gaussian broadcast channel is a group code for the single-user Gaussian channel in which a subgroup of the generating matrix group is used to partition the codewords into disjoint subsets called clouds. Distinct clouds represent distinct messages for the noisier channel while the individual codewords represent distinct messages for the other channel. The clouds share some of the distance properties of group codes. Necessary and sufficient conditions are given for finding good group codes for the Gaussian broadcast channel (in terms of minimum distance).