Sion's minimax theorem and Nash equilibrium of symmetric multi-person zero-sum game

About a symmetric multi-person zero-sum game we will show the following results. 1. The existence of a symmetric Nash equilibrium is proved by the Glicksberg fixed point theorem. 2. Sion's minimax theorem and the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium. 3. The existence of a symmetric Nash equilibrium is proved by Sion's minimax theorem and the coincidence of the maximin strategy and the minimax strategy. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. However, if it is symmetric, the maximin strategies and the minimax strategies constitute a Nash equilibrium. With only the minimax theorem there may exist an asymmetric equilibrium in a symmetric multi-person zero-sum game.