How do you find the generator for an additive group?

Question

I know that for a multiplicative group $G$, that $x \in G$ is a generator of $G$ iff all elements in $G$ are $x^{n}, n \in \mathbb{Z}$. Is this the same for an additive group? If not, why? What about the generators of groups under other operators?

Answer 1

Let $G$ be an additive group and $x\in G$. $x$ is a generator of $G$ if and only if we have : $$G=\{nx,n\in\mathbb{Z}\}.$$ For example, $\mathbb{Z}$ is an additive group generated by $1$.

Answer 2

If you are only looking at groups, then the name of the operation (addition, multiplication, etc.) does not matter...

If you write your group operation using "multiplication" ($x y$) then the definition of generator looks like "$x \in G$ is a generator for $G$ if every element in $G$ is of the form $x^n$ for some integer $n$."

If you write your group operation using "addition" ($x+y$), then the definition looks like "$x \in G$ is a generator for $G$ if every element of $G$ is of the form $nx$ for some integer $n$."

In complete generality, if the operation is expressed as $x*y$, then $x$ is a generator of $G$ if every element of $G$ is of the form $$\underbrace{x*x*\cdots*x}_{n}$$ for some integer $n$.