Find all generators of $\langle a\rangle , \langle b\rangle,$ and $\langle c\rangle$

Question

Suppose that $\langle a\rangle , \langle b\rangle,$ and $\langle c\rangle$ are cyclic groups of orders $6, 8,$ and $20,$ respectively.

Find all generators of $\langle a\rangle , \langle b\rangle,$ and $\langle c\rangle$.

Answer 1

Let's look at $\langle a\rangle$ which is a cyclic group of order $6$. Hence, $\langle a\rangle = \{a^0 = e, a^1, a^2 \cdots a^5\}$, by the definition of $\langle a\rangle$, where $|\langle a \rangle| = 6$.

Now, the elements of this group that are also generators of the group $\langle a \rangle$ are those elements $a^m$, where $m$ is an integer, $0\lt m \lt 6$, such that $\gcd(m, 6) = 1$. In this case, that would be only those $a^m$, where $m \in \{1, 5\}$: we already know $a^1 = a$ is a generator, but $a^5$ generates $\langle a\rangle$ as well.

Can you generalize this for a cyclic group of order $n$? You can apply this, then, in the same manner, to $\langle b\rangle$ and $\langle c\rangle$.

Answer 2

Hint: If $G=\langle z\rangle$ is a cyclic group of order $n,$ say, then you know that the elements of $G$ are (in multiplicative notation) $1,z,z^2,...,z^{n-1}.$ For which $k\in\{0,1,2,...,n-1\}$ does $z^k$ have order $n$?