posted by Bob .
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 timeswe must apply γ to the resulting set in order to guarantee the set is back in its original configuration when we stop?
Each permutation can be decomposed in terms of cyclical permutations. The GCM of the cycle lengths is the number of times you need to apply the permutation to get the same result back. This number is, of course, different for each permutation, so we need to find the LCM of these numbers taken over all the permutations.
This is then a LCM of a set of LCM of cycle lengths, so we can just consider all possible cycle lengths and take the LCM of these. This is clearly the product of all the prime numbers smaller or equal to 8, which is 2*3*5*7 = 210, so the least number of times you need to apply the permutation to be guaranteed that you to get the original configuration is 209.
but 209 is incorrect