## Mappings, Morphisms and all

As you may have noticed, the mappings are pretty important here.
So lets look closer.
Given a group G, I use the terminology G->G or better M(G) for
the mappings from G to itself.
Some of these mappings are special, in particular those
mappings that respect the addition on the group G.
A mapping f from M(G) respects the addition if
f(a+b) = f(a) + f(b) for all a,b in G and is
called an endomorphism.
We will use the terminology E(G) for the set of all
endomorphisms on a group.

Given two mappings f,g from M(G), we can define
their sum as a point-wise sum; that is
(f+g)(x) = g(x) + g(x). This is the usual
sum of functions that we use.
The other operation of interest is
composition. The composition of two
functions f,g from M(G) is (fog)(x) = f(g(x)).

Thus we come up with two operations on M(G).
If we have an abelian group, then the set E(G)
is closed under these two operations. That is,
the sum of two endomorphisms is an endomorphism,
and the composition of two endomorphisms is an endomorphism.
Thus the algebra (E(G), +, o) is a ring.

Similarly, if we take take a non abelian group, or in fact
any group at all, we can look at the collection
of all mappings on that group under pointwise addition and
composition and find that we have a near-ring (M(G),+,o).
This will be a right near-ring since
((f+g)oh)(x) = (f+g)(h(x)) = (f(h(x)) + g(h(x))) = (foh + goh)(x).
In general, the set of endomorphisms of a non-abelian
group is not closed. Thus we speak of the set of mappings
generated by the endomorphisms under the operations of
pointwise addition and composition, and still use the
terminology E(G) for this near-ring (E(G),+,o).