Page 42 - DMTH403_ABSTRACT_ALGEBRA

Unit 2: Groups

Hence, x + y > 0. Then Notes
2
2
 x  y 
(x,y).  2 2 , 2 2 
 x  y x  y 



= x . 2 x 2  y . ( y) 2 , x . 2  y 2  y 2 x 2  

2
 x  y x  y x  y x  y 
= (1, 0)
 x  y 
Thus,  2 y 2 , 2 y 2  is the multiplicative inverse of (x, y) in C.
 x  x  
Thus, (C, +) is a group and (C*,.) is a group. (As usual, C* denotes the set of non-zero complex
numbers.)
Now let us see what we have covered in this unit.

Self Assessment

1. For a binary operation * on S and (a, b)  S × S, we denote *(a, b) by ...............
(a) a * b (b) (a, b)*
(c) ab* (d) ba*

2. Binary operations associates a ............... of S to every ordered pair of elements of S.
(a) same element (b) different element
(c) unique element (d) single set element
3. Suppose their exist a, b  S such that S * a = e = a * s and S * b = e = b * s, e being the identity
element for *, then
(a) a = b (b) b = a
–1
(c) a = b (d) a = b
2
–1
4. For x  R, if b is inverse of x, the you should have b  x = 1. Then x is equal to
-1
(a) 2 – x (b) x – 2
(c) 2 x (d) 2 + x –1
–1
5. ............... are named after the gifted young Norwegian mathematician Niels Henrik Abel.
(a) Abelian group (b) Sub group
(c) Normal group (d) Cyclic group

2.5 Summary

Here we discussed various types of binary operations.

Also defined and given examples of groups.

We proved and used the cancellation laws and laws of indices for group elements.

In this unit we discussed the group of integers modulo n, the symmetric group and the

group of complex numbers.

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