Consider an abelian group $G$. I will define a new group $E$ whose elements are the set of all endomorphisms from $G$ to $G$.
The multiplication $*$ in $E$ should be defined like so:
$a*b = c$ where $c:G\rightarrow G$ is the map defined by $c(x) = a(x)+b(x)$
From here it is easy to see $E$ is a group, the identity is the map taking all elements to the identity of $G$. The inverse of a map $a$ is a map $b$ that takes $x$ to $-a(x)$. Also it is easy to verify associativity, thus $E$ is a group.
I would like to know what this group $E$ is called in relation to the group $G$. Is it called the endomorphism group? If not what is its name?
Thanks.