Let $G$ be the group of units of the commutative group $\mathbb{Z}/2007\mathbb{Z}$ and consider the group homomorphism $h\colon G\to G$ where $h(g)=g^n$ for $n\ge1$.
How can I find the order of $\ker h$?
Let $G$ be the group of units of the commutative group $\mathbb{Z}/2007\mathbb{Z}$ and consider the group homomorphism $h\colon G\to G$ where $h(g)=g^n$ for $n\ge1$.
How can I find the order of $\ker h$?
The kernel of $h$ is the elements satisfying $g^n=1$. Do you know how to find the number of elements of a given order in a cyclic group? Somewhere along the way, you will have to also find the factorization of 2007.
EDIT: By the way, I'm a little worried about your phrase, "the group of units of the commutative group ${\bf Z}/2007{\bf Z}$." ${\bf Z}/2007{\bf Z}$ is a commutative group under addition and, as a group, all its elements are units. But it seems that what you really mean is "the group of units of the commutative ring ${\bf Z}/2007{\bf Z}$." Those would be the elements that have multiplicative inverses, and that is what we are discussing in the comments.