Groups - Elliptic Curves: Number Theory and Cryptography - page 476

THEOREM B.3
A finite abelian group is isomorphic to a group of the form

Z n 1 ⊕ Z n 2 ⊕···⊕ Z n s

with n i |n i +1 for i =1 , 2 ,...,s− 1 . The integers n i are uniquely determined
by G.

An abelian group G is called finitely generated if there is a finite set
{g 1 ,g 2 ,...,g k } contained in G such that every element of G can be written
(not necessarily uniquely) in the form

m 1 g 1 + ··· + m k g k

with m i ∈ Z .

THEOREM B.4
A finitely generated abelian group is isomorphic to a group of the form

Z r ⊕ Z n 1 ⊕ Z n 2 ⊕···⊕ Z n s

with r ≥ 0 and with n i |n i +1 for i =1 , 2 ,...,s− 1 . The integers r and n i are
uniquely determined by G.

The subgroup of G isomorphic to

Z n 1 ⊕ Z n 2 ⊕···⊕ Z n s

is called the torsion subgroup of G . The integer r is called the rank of G .
This theorem can be used to prove the following.

THEOREM B.5
Let G 1 ⊆ G 3 be groups and assume that, for some integer r,bothG 1

and G 2 are isomorphic to Z r .ThenG 2 is isomorphic to Z r .

G 2 ⊆

For example, G 1 =12 Z , G 2 =6 Z ,and G 3 = Z , each of which is isomorphic
as a group to Z , satisfy the theorem. This theorem is used in the text when
G 1 and G 3 are lattices in C .Then G 1 and G 3 are isomorphic to Z 2 . f
G 1 ⊆ G 2 ⊆ G 3 ,then G 2 Z 2 , so there exist ω 1 , ω 2 such that G 2 = Z ω 1 + Z ω 2 .
Since G 1 is a lattice, it contains two vectors that are linearly independent over
R .Since G 1 ⊆ G 2 ,thisimpliesthat ω 1 and ω 2 are linearly independent over
R . Therefore, G 2 is a lattice.