Who has considered the LCM of the multiplicative order of the divisors of p-1, where p is a prime, for which p-1 is square free?

Question

A colleague of mine and I have been considering the behavior of divisors of p-1 with respect to multiplicative order in Z/(p), for primes p with p-1 square free. The least common-multiple of the orders of the divisors of p-1, which we call L(p), crops up naturally, and we are hoping someone can give references to this concept. In particular, we would like to know whether or not it is known if it is true that L(p)= p-1, whenever there is no divisor of p-1 whose order is L(p).