Note that if f and g are bijections from S to S, that h = g ?f is also a bijection from S to S. Suppose you want to write a program capable of computing the result of any bijection on S - given an element of S and a bijection, the program outputs what that element is mapped to. One approach to this would be to write a program for each possible bijection, and simply run the correct program for the desired bijection.

However: note that if we have a program for g and a program for f, we do not need a program explicitly for h - we could simply run the program to compute f, then the program to compute g, feeding the output of the ?rst in as the input to the second. In this way, we say that f and g generate h. By composing more functions / chaining more programs, we can generate other functions / programs as well.

Find a minimal set of generators that will allow you to compute any bijection from S to S.