Groups - Elliptic Curves: Number Theory and Cryptography - page 477

Homomorphisms

Let G 1 , G 2 be groups. A homomorphism from G 1 to G 2 is a map ψ :
G 1 → G 2 such that ψ ( g + h )= ψ ( g )+ ψ ( h ) for all g, h ∈ G 1 . In other words,
the map takes sums in G 1 to the corresponding sums in G 2 .The kernel of
ψ is

Ker ψ = { g ∈ G 1 | ψ ( g )=0 } .

The image of ψ is denoted ψ ( G 1 ), which is a subgroup of G 2 . The main result
we need is the following.

THEOREM B.6
Assume G 1 is a finite group and ψ : G 1 → G 2 is a homomorphism. Then

# G 1 =(#Ker ψ )(# ψ ( G 1 )) .

In fact, in terms of quotient groups, G 1 / Ker ψ ψ ( G 1 ).