Examples on isomorphism and homomorphism Could someone please explain to me how isomorphisms and homomorphisms work?
For an isomorphism, I know we need to follow the following four steps: 
$\ \ \ $1) define a candidate, 
$\ \ \ $2) show it is $1$-$1$ 
$\ \ \ $3) show it is onto 
$\ \ \ $4) show it is closed.
But, I don't know how to actually apply it if we are given a question like what should I be focusing on.
And the same for homomorphisms.
Any help with examples would be appreciated. Thank you.
 A: For an isomorphism take the symmetries of a regular pentagon, and the symmetries of the five-pointed star obtained by joining alternate vertices of the pentagon. Every symmetry of the pentagon converts into a symmetry of the star, and vice-versa. Although the two figures look different their symmetries are the same. That's an isomorphism.
For a homomorphism consider the integers and split them into even and odd numbers as if whether they are even or odd is all that matters. We have E+E = O+O = ExE = OxE = ExO = E; and O+E = E+O = OxO = O. We have abstracted a feature of these numbers (oddness or evenness) in such a way that it respects the original arithmetic - the multiplication and addition. That's a homomorphism.
A: Let $(A,\triangle)$ and $(B,\triangledown)$ be groups. $f:A\rightarrow B$ is a homomorphism if $\forall x,y\in A$ 
$$ f(x\triangle y) = f(x)\triangledown f(y) $$
This is basically saying that $f$ preserves the group structure when it maps element of $A$ to elements of $B$. Homomorphisms tell us about the similarities between two groups.
For example, take the integers under addition modulo 2 and 4, $\mathbb Z_2$ and $\mathbb Z_4$. Let $f:\mathbb Z_2 \rightarrow \{0,2\} \subset \mathbb Z_4$ such that $f(0)=0$ and $f(1) = 2$. $f$ is a homomorphism (note that it is injective). Since the two groups are abelian it suffices to show that
$$\begin{align}
&f(0+_20) = f(0)+_4f(0) = 0+_40 = 0 = f(0) \\
&f(0+_21) = f(0)+_4f(1) = 0+_42 = 2 = f(1) \\
&f(1+_21) = f(1)+_4f(1) = 2+_42 = 0 = f(0)
\end{align}$$
This tells us that the structure $\mathbb Z_2$ is equivalent to the structure of the subgroup $\{0,2\}$ of $\mathbb Z_4$.
It is important to note that homomorphisms fix the identity element. If $e$ is the identity element in $A$ and $x \in A$ then $f(x\triangle e) = f(x)\triangledown f(e)$ and $f(x\triangle e) = f(x)$ (simplifying inside of $f$). So, $f(e)$ must be the identity element for $B$.
An isomorphism is a bijective homomorphism. If two groups are isomorphic they have the same group structure.
An epimorphism is a surjective homomorphism. If there is an epimorphism from $A$ to $B$, it implies that $B$ is isomorphic to some quotient group of $A$.
A monomorphism is an injective homomorphism. A monomorphism from $A$ to $B$ implies that $A$ is isomorphic to some subgroup of $B$ (as we saw above).
