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

摘要

For each positive integer n, let U(Z/nZ) 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(Z/nZ) 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,