  We consider the relation between Sion's minimax theorem for a continuous
function and a Nash equilibrium in a five-players game with two groups which is
zero-sum and symmetric in each group. We will show the following results.
  1. The existence of Nash equilibrium which is symmetric in each group implies
Sion's minimax theorem for a pair of playes in each group. 2. Sion's minimax
theorem for a pair of playes in each group imply the existence of a Nash
equilibrium which is symmetric in each group.
  Thus, they are equivalent. An example of such a game is a relative profit
maximization game in each group under oligopoly with two groups such that firms
in each group have the same cost functions and maximize their relative profits
in each group, and the demand functions are symmetric for the firms in each
group.
