Groups - Elliptic Curves: Number Theory and Cryptography - page 474

Appendix B

Groups

Basic definitions

Since most of the groups in this topic are additive abelian groups, we'll
use additive notation for the group operations in this appendix. Therefore, a
group G has a binary operation + that is associative. There is an additive
identity that we'll call 0 satisfying

0+ g = g +0= g

for all g ∈ G .Each g ∈ G isassumedtohaveanadditiveinverse − g satisfying

( −g )+ g = g +( −g )=0 .

If n is a positive integer, we let

ng = g + g + ··· + g ( n summands) .

If n< 0, we let ng = − ( |n|g )= − ( g + ··· + g ).
Almost all of the groups in this topic are abelian, which means that g + h =
h + g for all g, h ∈ G .
If G is a finite group, the order of G is the number of elements in G .The
order of an element g ∈ G is the smallest integer k> 0 such that kg =0.
If k is the order of g ,then

ig = jg ⇐⇒ i ≡ j (mod k ) .

The basic result about orders is the following.

THEOREM B.1 (Lagrange's Theorem)
Let G be a finite group.

1. Let H beasubgroupofG. Then the order of H divides the order of G.

2. Let g ∈ G. Then the order of g divides the order of G.