Page 34 - Advanced Linear Algebra

18 Advanced Linear Algebra

(
inverses. However, the set of invertible matrices of size d is a nonabelian)
group under multiplication.
A group . is finite if it contains only a finite number of elements. The
cardinality of a finite group is called its order and is denoted by ² . ³ or
.
.
simply (( . Thus, for example, { ~ ¸ Á Á Ã Á c ¹ is a finite group under
addition modulo , but C ² s Á ³ is not finite.

Definition A subgroup of a group is a nonempty subset of that is a
.
.
:
group in its own right, using the same operations as defined on .
.
Cyclic Groups
If is a formal symbol, we can define a group to be the set of all integral
.

powers of :

.~ ¸ “  ¹
{
where the product is defined by the formal rules of exponents:

~ b
This group is denoted by º » and called the cyclic group generated by . The

identity of º » is ~ . In general, a group . is cyclic if it has the form
.~ º » for some  ..

We can also create a finite group *² ³ of arbitrary positive order by

declaring that ~ . Thus,

* ² ³ ~ ¸ ~ Á Á Á Ã Á c ¹

where the product is defined by the formal rules of exponents, followed by
reduction modulo :

~ ² b ³ mod
This defines a group of order , called a cyclic group of order . The inverse

of is ² c ³ mod .

Rings
Definition A ring is a nonempty set , together with two binary operations,
9
called addition denoted by b ) and multiplication denoted by juxtaposition ,
)
(
(
for which the following hold:
)
1 9 is an abelian group under addition
2 )(Associativity ) For all Á Á  9 ,
² ³ ~ ² ³