Page 41 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes

Note In the notation of a cycle, we don’t mention the elements that are left fixed
 2 5
by the permutation. Similarly, the permutation  5 3  is the cycle (1 2 5 3 4 ) in S .
5
 
Now let us see how we calculate the composition of two permutations. Consider the following
example in S .
5

 1 2 3 4 5   1 2 3 4 5 

 °  =  2 5 4 3 1   5 3 4 1 2 
   
 1 2 3 4 5 
=   (1) (2) (3) (4) (5)  

 1 2 3 4 5 
=   (5) (3) (4) (1) (2)  

 1 2 3 4 5
=  1 4 3 2 5   (2 4).
 
since 1, 3 and 4 are left fixed.
And now let us talk of a group that you may be familiar with, without knowing that it is a
group.

2.4.3 Complex Numbers

In this sub-section we will show that the set of complex numbers forms a group with respect to
addition. Some of you may not be acquainted with some basic properties or complex numbers.

Consider the set C of all ordered pairs (x, y) of real numbers. i.e.. we take C = R × R. Define
addition (+) and multiplication (.) in C as follows:
(x , y ) + (x , y ) = (x + x , y + y ) and
1
2
2
2
1
1
2
1
(x , Y ) . (x , y ) = (x x – y y , x y + x y )
1 2
2 1
2
1 2
1
1
2
2
1
for (x . y ) and (x , y ) in C.
2
1
1
2
This gives us an algebraic system (C, +, .) called the system of complex numbers. We must
remember that two complex numbers (x , y ) and (x , y ) are equal iff x = x and y = y .
2
2
1
2
1
2
1
1
You can verify that + and . are commutative and associative.
Moreover,
(i) (0, 0) is the additive identity.
(ii) for (x, y) in C, (–x, –y) is its additive inverse.
(iii) (1, 0) is the multiplicative identity.
(iv) if (x, y)  (0, 0) in C, then either x > 0 or y > 0.
2
2
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