Page 29 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Notes
b
n = – R.
a 2 b 2
Hence C is a multiplicative group.

Self Assessment

4. Show that the set of all odd integers with addition as operation is not a group.
5. Verify that the totality of all positive rationals form a group under the composition
defined by
a o b = ab/2
6. Show that the set of all numbers cos + i sin forms an infinite abelian group with respect
to ordinary multiplication; where runs over all rational numbers.

Composition (Operation) Table

A binary operation in a finite set can completely be described by means of a table. This table is
known as composition table. The composition table helps us to verify most of the properties
satisfied by the binary operations.
This table can be formed as follows:

(i) Write the elements of the set (which are finite in number) in a row as well as in a column.
(ii) Write the element associated to the ordered pair (a , a ) at the intersection of the row
i j
th
headed by a and the column headed by a . Thus (i entry on the left). (j entry on the top)
th
i j
= entry where the i row and j column intersect.
th
th
For example, the composition table for the group {0, 1, 2, 3, 4} for the operation of addition is
given below:
0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 5
2 2 3 4 5 6
3 3 4 5 6 7
4 4 5 6 7 8

In the above example, the first element of the first row in the body of the table, 0 is obtained by
adding the first element 0 of head row and the first element 0 of the head column. Similarly the
third element of 4 row (5) is obtained by adding the third element 2 of the head row and the
th
fourth element of the head column and so on.
An operation represented by the composition table will be binary, if every entry of the
composition table belongs to the given set. It is to be noted that composition table contains all
possible combinations of two elements of the set will respect to the operation.
Notes:

(i) It should be noted that the elements of the set should be written in the same order both in
top border and left border of the table, while preparing the composition table.

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