We were asked in my class to find all elements of $\langle a\rangle$ for all $a$ in $\Bbb Z / 48 \Bbb Z$.
The answer can be found here, and I understand how the cyclic groups are formed. Rather than tediously writing out all 48 cyclic groups, I am wondering what the pattern is, if there is one. I see that the group $\Bbb Z/48\Bbb Z$ is generated by $\langle1\rangle,\langle5\rangle,\langle7\rangle,\langle11\rangle,\langle13\rangle,\langle17\rangle,\langle19\rangle,\langle23\rangle,\langle25\rangle$, etc.
I know that $\langle a\rangle=\langle a^{-1}\rangle$, so $\langle 1\rangle = \langle 47\rangle, \langle 2\rangle = \langle 46\rangle…$
I feel like a pattern is glaring right at me, but I would love to hear from others! Thanks.
Best Answer
It all depends on whether the number is coprime to $48$. Any number coprime to $48$ will generate the whole group. Any number not coprime to $48$ will generate only multiples of the $\gcd$ of the number and $48$.