Basic Algebra - Computer Graphics and Geometric Modeling - page 825

abd

n +=,

where

a += +, b kn d

and k and d are integers with 0 £ d < n. It is easy to check that ( Z n ,+ n ) (or Z n for short)
is an abelian group called the group of integers modulo n (or mod n ).

B.4.4. Example. Let p be a prime number and let X = {1,2,...,p-1}. Define an oper-
ation · on X as follows: Let a, b Œ X . Choose integers k and r so that

ab =+ kp r , where 0 << r p .

n

and set

ab r ◊ =.

It is easy to check that ( X ,·) is a well-defined abelian multiplicative group. Note that
since any integer in X is relatively prime to p, Theorem B.1.1 implies that it has an
inverse with respect to ·.

Note. In the future, when dealing with a group (G,·) for which the operation · is
obvious from the context, we shall simply refer to “the group G.”

In the case of a general group one typically thinks of the group operation as a
multiplication operation and uses “1” as the group identity. The trivial groups con-
sisting of only one element will be denoted by 1 . One also simply writes “gh” for a
product of group elements g and h rather than “g · h”. If g is a group element, then g k
will denote the element

gg ◊ ◊ ◊◊◊ ◊

12

g

443 44

k times

In the abelian group case, the standard convention is to use additive notation, so that
we shall use “+” for the group operation, “0” will be the (additive) identity, and “-g”
will denote the inverse of g. The trivial abelian groups consisting of only one element
will be denoted by 0 . If k ΠZ , then kg will denote the element

gg + + ◊◊◊+ g

12 443 44

k times

Definition. Let (G,·) and (H,·¢) be groups. We say that (H,·¢) is a subgroup of
(G,·) provided that H Õ G and ·¢ agrees with ·, that is, ·¢=· | (H ¥ H).

B.4.5. Example. Two trivial subgroups of any group are the subgroup consisting of
only the identity and the whole group itself.