Find the generator of $(Z,*)$

Question

let $Z$ be the set of integers

let $\ast\ $ be an operation defined on $Z$ by $a*b=a+b-1$ $\forall \ $ $a,b$ $\in \ $$Z$

Is $(Z,*)$ cyclic if so find the generator of $(Z,*)$.

I think it is not cyclic group since if it is $1$ its inverse same as $1$ therefore there is no generator but I am not sure if I am wrong correct me.

Answer 1

Since $1$ is the identity element of $(\Bbb Z,*)$, then it is natural that it is not a generator. However, $2$ is a generator of $(\Bbb Z,*)$. To see why, see that $2*2=3$, $2*3=4$, $2*4=5$ and so on…