We extend the notion of morphism of monoids to morphism of groups. Let G and G' be two groups.
Definition
f : G -> G'
A morphism (of groups) from G to G' is a map
with the following properties:
- f(e) = e' (the unit element of G');
-
for all a,b
G, we have
f(ab) = f(a)f(b); -
for all a
G : f(a-1) = f(a)-1.
If two groups are isomorphic, i.e., if there exists an isomorphism from one to the other, then they have the same `shape' (morph=shape).
A morphism between two groups is also a morphism of the corresponding monoids.
The reverse is also true.
Proposition
f(gh) = f(g)f(h).
A map f : G -> G' is a morphism of groups
if and only if, for all g, h G,