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The antenna subset selection technique balances the performance and hardware cost in the multiple-input multiple-output (MIMO) systems, and the problems on the antenna selection in MIMO relay systems have not been fully solved. This paper considers antenna selection on amplify-and-forward (AF) and decode-and-forward (DF) MIMO relay systems to maximise capacity. Since the optimal antenna selection algorithm has high complexity, two fast algorithms are proposed. The selection criterion of the algorithm for AF relay is to maximise a lower bound of the capacity, but not the exact capacity. This criterion reduces algorithmic complexity. The algorithm for DF relay is an extension of an existing antenna subset selection algorithm for one-hop MIMO systems. The authors show the derivations of the algorithms in detail, and analyse their complexity in terms of numbers of complex multiplications. Simulation results show that the proposed algorithms for both cases achieve comparable performance to the optimal algorithm under various conditions, and have decreased complexity.