The Number of Non-cyclic Sylow Subgroups of the Multiplicative Group Modulo n

Abstract

AbstractFor each positive integer n, let $U(\mathbf {Z}/n\mathbf {Z})$ denote the group of units modulo n, which has order $\phi (n)$ (Euler’s function) and exponent $\lambda (n)$ (Carmichael’s function). The ratio $\phi (n)/\lambda (n)$ is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of $U(\mathbf {Z}/n\mathbf {Z})$ is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants $C_1, C_2>0$, $$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$ for all n from a sequence of asymptotic density 1. We sharpen their result by showing that W(n) has normal order $\log \log {n}/\log \log \log {n}$.