Homomorphisms are functions which satisfy:

f(g1)f(g2)=f(g1g2)

In general, homomorphisms are many-to-one.

Example:
let N a normal subgroup of G
for G'=G/N
The map G -> G' defined by g -> gN is a homomorphism. f(g)=gN forall g in G
forall g1,g2 in G g1Ng2N=g1g2N.

Theorem:
Let K=kernel of homomorphism f:G -> G'
G/K ~= G' (G/K isomorphic to G'

proof:
We need Pi:f(g) -> gK an isomorphism.
Pi one-to-one => f(g1) = f(g2) <=> g1K=g2K <=> f(g1)f(g2)^=f(g1g2^)=e' <=> g1g2^ in K <=> Kg1=Kg2