Difficult# Number of Roots of Unity in an Abelian Group

ABSALG-XZWEJ4

In answering this question, you may use the following ** true** facts:

**Fact 1** - If $A$ is a finite abelian group and $H$ is the largest cyclic subgroup of $A$, then:

the order of any other cyclic subgroup of $A$ divides the order of $H$.

**Fact 2** - In the group $A=({\mathbb Z}/p{\mathbb Z})^\ast$, $p\ge 2$ prime, there are:

at most $m$ solutions to $x^m=1$, for every $m\ge 1$.

**Which of the following is FALSE**?

(Notation: For $N\ge 2$ an integer, we denote by $({\mathbb Z}/N{\mathbb Z})^\ast$ the group of integers mod $N$ coprime to $N$.)