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 determine the minimum number of additional times we must apply γ to the resulting set in order to guarantee that the set is back in its original configuration, we need to calculate the order of the permutation γ.
The order of a permutation is defined as the smallest positive integer k such that γ^k = e, where e represents the identity permutation (the permutation that does not change the elements of the set).
To find the order of a permutation, we can keep applying the permutation until we obtain the identity permutation. Each time we apply the permutation, we keep track of the number of times we have applied it. Once we obtain the identity permutation, the number of times we applied the permutation is equal to the order of the permutation.
In this case, since γ is a permutation on eight elements, we can apply γ repeatedly until we obtain the identity permutation. We start by applying γ to the set and count the number of times we apply it until we reach the identity permutation.
Note that since γ is not specified, we cannot determine the exact number of times we need to apply it without further information. Therefore, it is not possible to determine the minimum number of additional times we must apply γ to guarantee the set's original configuration without knowing the specific permutation γ being used.