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Abelian group Algebraic structures

(Monday, May 3, 2010)



A group áG, *ñ is said to be an Abelian group if the commutative law with respect to the binary operation * hold in G, i.e., for all a, b e G, a * b = b*a.
Order of a group
A natural characteristic of a group
áG, *ñ is the number of elements it contains. This is called the order of the group.    
If the group áG, *ñ be such that the set G contains n elements , n e N, then it is called a finite group of order n.  If G is an infinite set, then the group áG, *ñ will be called an infinite group.
Example
 
1. The set of integers Z is a group with
    respect to the operation of addition.
    (a) Closure property holds in Z, the
          sum of any to integers being an
          integer.
    (b) Associative law will holds for any
          three integers x, y, z e Z for
                (x + y) + z = x + (y + z)
    (c) The integer 0 is the identity
          element since a + 0 = 0 + a,
          where a is an integer
    (b) The inverse of any element a e Z
          is its negative, i.e., -a e Z, since
          a + (-a) = (-a) + a = 0.
Hence, áZ, *ñ is a group. Moreover, as
a + b = b + a, for all a, b e Z, it is an Abelian group.

2. The Q0 be the set of all non-zero rational numbers, then the algebraic structure áQ0, .ñ forms a multiplicative Abelian group.
G1. For all a, b e Q0 Þa.b e Q0
G2. (a.b).c = a.(b.c) for all a, b, c e Q0
G3. 1 e Q0 is the identity element,
       a.1 = 1.a = a, for a e Q0

G4. For all a e Q0, 1/a or a-1 e Q0, such that a.1/a = 1/a.a = 1. 1/a or a-1 is called the inverse of a.
Moreover, for all a, b e Q0, a.b = b.a holds good. Hence, áQ0, .ñ is a multiplicative Abelian group.

Note: The set of all rational numbers do not from a group with respect to the binary operation of multiplication.  For the rational number 0 has no multiplicative inverse.
 

 




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