We wish to select, from k(≥;2) independent normal populations, the one associated with the largest mean, assuming that the common variance is unknown. We adopt the “indifference zone” approach of Bechhofer (1954), and first consider the sequential selection rule of Robbins et al. (1968). We then turn our attention to the “permutation-invariant” idea, as introduced in Mukhopadhyay et al. (1989). Often, sequential stopping rules, such as the stopping rule of Robbins et al. (1968)) are dependent on the order in which the data is observed. With permutation-invariant stopping rules, all possible permutations of the data are subjected to the stopping rule. We apply this idea to the stopping rule of Robbins et al. (1968)) and develop a permutation-invariant version. First-order asymptotic results are established, and compared to the corresponding results for the non-permutation-invariant procedure. Computer simulations are used to compare the moderate sample size performances of the two procedures.