Estimating the density of a set of primes with applications to group theory

We estimate the asymptotic density of the set $\bar{A}$ of primes $p$ satisfying the constraint that $p+1$ and $p-1$ have only one prime divisor larger than $3$. We also estimate the density of a maximal subset $\bar{B} \subset \bar{A}$suchthatfor$p_1, p_2 \in \bar{B}$ no common prime divisor of $p_1(p_1+1)(p_1-1)$ and $p_2(p_2+1)(p_2-1)$ is larger than $3$. Assuming a generalized Hardy–Littlewood conjecture, we prove that for both $\bar{A}$ and $\bar{B}$ the number of elements lesser than $x$ is asymptotically equal to a constant times $x\mathrm{/}(\log x{)}^3$.