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# Number of Cyclic Subgroups, Reduced Residues

ABSALG-KN10XE

Let $G$ be a group. The order of $G$ is the number of elements in the set $G$.

The cyclic subgroups of $G$ are the subgroups equal to:

$\langle g\rangle:=\{g^n\mid n\in{\mathbb Z}\}$ (multiplicative notation)

...for some $g\in G$.

Let $({\mathbb Z}/m{\mathbb Z})^\ast$ denote the group of residue classes mod $m$ of the integers coprime to $m$ (i.e. the reduced residues).

Which of the following is TRUE?

A

$({\mathbb Z}/12{\mathbb Z})^\ast$ has exactly $1$ cyclic subgroup of order $2$

B

$({\mathbb Z}/12{\mathbb Z})^\ast$ has exactly $2$ cyclic subgroups of order $2$

C

$({\mathbb Z}/18{\mathbb Z})^\ast$ has exactly $2$ cyclic subgroups of order $6$; exactly $2$ cyclic subgroups of order $3$; exactly $1$ cyclic subgroup of order $2$; and exactly $1$ cyclic subgroup of order $1$

D

$({\mathbb Z}/18{\mathbb Z})^\ast$ has exactly $1$ cyclic subgroup of order $6$; exactly $1$ cyclic subgroup of order $3$; exactly $1$ cyclic subgroup of order $2$; and exactly $1$ cyclic subgroup of order $1$

E

$({\mathbb Z}/18{\mathbb Z})^\ast$ has exactly $1$ cyclic subgroup of order $9$; exactly $1$ cyclic subgroup of order $6$; exactly $1$ cyclic subgroup of order $3$; exactly $1$ cyclic subgroup of order $2$; and exactly $1$ cyclic subgroup of order $1$