γ is a permutation on eight elements, though you are not told which permutation it is. If γ is applied to an 8-element set, what is the minimum number of additional times we must apply γ to the resulting set in order to guarantee the set is back in its original configuration when we stop?

To find the minimum number of additional times we must apply γ to the resulting set in order to guarantee the set is back in its original configuration, we need to determine the order of the permutation γ.

The order of a permutation is the smallest positive integer n such that applying the permutation n times will result in the original configuration. In other words, it is the smallest number of repetitions needed to return to the starting point.

To find the order of the permutation γ, we can apply γ repeatedly until we reach the original configuration. We start with the initial set, apply γ once, and continue applying γ until the set returns to its original configuration.

So, to find the minimum number of additional times we must apply γ, we need to find the order of the permutation γ.