A well-known result about Cyclic Groups is:
Let $G = \langle a \rangle$ be a cyclic group of order $n$. Then, for every $k \in \mathbb{Z}$:
\begin{align}
G = \langle a^k \rangle \Leftrightarrow \gcd(n,k) = 1.
\end{align}
In order to illustrate this result my textbook considers the cyclic group $\mathbb{Z}/12\mathbb{Z} = \{\overline{0}, \overline{1}, \ldots, \overline{11}\}$ and states that the generators of this group are $\overline{1}, \overline{5}, \overline{7}, \overline{11}$.
My question now is: How do I have to understand the expression $a^k$?
Usually, I would read it as $a^k = \underbrace{a \cdot a \cdot \ldots \cdot a}_{k-\text{times}}$, but in the context of the above mentioned example it would rather be $a^k = \underbrace{a + a + \ldots + a}_{k-\text{times}}$. (Using "$\cdot$" would not make much sense since $\overline{1}$ for example is no generator.)
Since I confused myself a little bit about this, there is a more general question: If we are talking about Cyclic Groups do I always have to think about "$+$" as the operation instead of "$\cdot$"? Just to make sure: When I am talking about $(\mathbb{Z}/n\mathbb{Z})^{\times}$ (the units of $(\mathbb{Z}/n\mathbb{Z})$) this is always a group with "$\cdot$", right?
Best Answer
If $(G, *)$ is a group and $a \in G$, then $a^k$ simply means $\underbrace{a*\cdots * a}_{k \text{ times}}$ for $k > 0$.
If you are denoting the operation as $+$, then you have $+$ instead of $*$ above.
So it simply is a matter of what symbol you use for the operation.
Now, given an abelian group, it is a common convention to use $+$ for the group operation since it's commutative. In this case, one may also sometimes write $ka$ instead of $a^k$.
So, if your group is cyclic, then it indeed abelian and you may identify it with $\Bbb Z/n\Bbb Z$ and use $+$.
However, one also has another very natural cyclic group which is written multiplicatively, namely, the group of $n$-th complex roots of unity.
This is the group $\{1, \zeta, \ldots, \zeta^{n - 1}\}$, where $\zeta = e^{2\pi\iota/n}.$
To summarise: Depending on your symbol for the group operation, you have the appropriate definition of $a^k$.
So you are correct that you would think of "$\cdot$" when thinking about $(\Bbb Z/n\Bbb Z)^\times$.