Properties of large prime divisors of numbers of the form $p-1$

The main result of this paper is the fact that the fraction of primes $p\le x$ satisfying the condition that $p-1$ has a prime divisor $q>\exp(\ln x/\ln\ln x)$ and the number of prime divisors of $q-1$ essentially differ from $\ln\ln(x/n)$, where $n=(p-1)/q$, tends to zero as $x$ increases.