Group Mappings
Definition 1.5 – Homomorphism
A homomorphism is a mapping $\mu : G\to H$ between groups $G$ and $H$ which is injective or surjective. In other words, if $g_{a}\cdot g_{b} = g_{c}$, then $\mu(g_{a}) \cdot \mu(g_{b}) = \mu(g_{c})$.
There exists at least one homorphism between any group $G$ and $H$, the one which maps all elements of $G$ to the identity of $H$.
An isomorphism is a mapping $\mu: G \to H$ between groups $G$ and $H$ which is bijective. Simply put, the groups do the same thing.
Definition 1.6 – Isomorphism
If there exists an isomorphism $\mu: G \to H$, then there necessarily exists an inverse mapping $\mu: H \to G$. Such groups are called isomorphs ($G \cong H$).
Compared to homomorphisms, isomorphs are strong relations between groups.
Mappings
A mapping (application in French) is the association of all the elements from some set $X$, to some or all elements of another set $Y$.
Injection
If the mapping creates distinct associations to some, but not all, elements of $Y$, the mapping is called an injection (or one-to-one).
Surjection
If the mapping creates non-distinct associations to all elements of $Y$, the mapping is called a surjection (or onto).
Bijection
If the mapping creates distinct associations to all elements of $Y$, the mapping is called a bijection (or one-to-one and onto).