Finite Group
•If a
group has a finite number of elements, it is referred to as a finite
group, and
the order
of
the group is equal to the number of elements in the group.
•Otherwise, the
group is an infinite group.
Abelian Group
•A
group is said to be abelian if it satisfies the following additional
condition:
(A5)
Commutative: a Ÿb
= b
Ÿa for all a, b in G.
•If
the operation on the set elements is commutative, the group is called an abelian
group.
•The set of
integers (positive, negative, and 0) under addition is an
abelian
group.
•The set of
nonzero real numbers under multiplication is an abelian group.
•The set Sn from the preceding example is a group but
not an abelian
group for n
>
2.
•When the
group operation is addition, the identity element is 0; the inverse element of a
is –a; and
subtraction is defined with the following rule: a - b = a + (-b).
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