Could someone help me with this question? I am a little stuck...
Find all generators of each subgroup of order $8$ in $\Bbb{Z}_{32}$.
By the fundamental theorem of cyclic groups, there is exactly one subgroup $<32/8>=<4>$ of order $8$. Then $<4>=\left \{ 0,4,8,12,16,20,24,28 \right \}$. However, I am not sure how to find the generators for $<4>$. Are they the numbers $x$ in $<4>$ such that $gcd(x,32)=1$?
$Hint$ : Try to find the no. of subgroups of order $8$. Generally in a finite ordered cyclic group, no. of subgroups of order $m$ is $φ(m)$.