$Z$ is cyclic and has generators 1 and -1 I know that group G is cyclic if there exist $ g \in G$ such that $ G = \{g^k : k \in Z\}$. However I don't understand how Z has generators $1$ and $-1$. 
Does $g^k$, so in this case, $1^k$ mean doing "$1$" $k$ times? Like adding $1$ $k$ times? And not $1$ to the power of $k$?
I think my confusion might be in the notation.
Thanks
 A: If $G$ is a group with operation $*$, that $g$ is a generator means that every element can be written as
$$ g *g * \cdots * g $$
or is the inverse of such a thing, or is the identity.
When the group operation is written as multiplication, we abbreviate
$$ \underbrace{g *g * \cdots * g}_{n\;g\text{s}} =
\underbrace{gg\cdots g}_{n\;g\text{s}} = g^n  \qquad \text{and }g^0=e,\;\;g^{-n} = (g^n)^{-1}$$
However, when a group whose operation is written with a $+$ symbol the abbreviation we use instead is
$$ \underbrace{g *g * \cdots * g}_{n\;g\text{s}} =
\underbrace{g +g + \cdots + g}_{n\;g\text{s}}
= ng  \qquad \text{and }0g=e,\;\;(-n)g = -(ng)$$
but from a group-theoretic viewpoint it's really the same thing that's going on, just notated differently.
When your definition of cyclic group speaks about $g^k$ it happens to be using multiplicative notation. When you want to apply to this definition to $(\mathbb Z,{+})$ you need to mentally recognize $g^k$ an abbreviation for a repeated addition, so what it actually says is that $\mathbb Z$ is cyclic because every element of $Z$ can be written as $1+1+\cdots+1$ or is the identity ($0$) or is the negative of $1+1+\cdots+1$.
