1 results
The Number of Non-cyclic Sylow Subgroups of the Multiplicative Group Modulo n
- Part of
-
- Journal:
- Canadian Mathematical Bulletin / Volume 64 / Issue 1 / March 2021
- Published online by Cambridge University Press:
- 08 June 2020, pp. 204-215
- Print publication:
- March 2021
-
- Article
- Export citation
-
For 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 
$C_1, C_2>0$, for all n from a sequence of asymptotic density 1. We sharpen their result by showing that W(n) has normal order
$$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$
$\log \log {n}/\log \log \log {n}$.
