Page 30 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes (ii) Generally a table which defines a binary operation ‘.’ on a set is called multiplication table,
when the operation is ‘+’ the table is called an addition table.

Group Tables

The composition tables are useful in examining the following axioms in the manner explained
below:
1. Closure Property: If all the elements of the table belong to the set G (say) then G is closed
under the Composition o (say). If any of the elements of the table does not belong to the
set, the set is not closed.
2. Existence of Identity: The element (in the vertical column) to the left of the row identical
to the top row (border row) is called an identity element in the G with respect to operation ‘o’.
3. Existence of Inverse: If we mark the identity elements in the table then the element at the
top of the column passing through the identity element is the inverse of the element in the
extreme left of the row passing through the identity element and vice versa.
4. Commutativity: If the table is such that the entries in every row coincide with the
corresponding entries in the corresponding column i.e., the composition table is
symmetrical about the principal or main diagonal, the composition is said to have satisfied
the commutative axiom otherwise it is not commutative.
The process will be more clear with the help of following illustrative examples.

Illustrative Examples

Example 8: Prove that the set of cube roots of unity is an abelian finite group with respect
to multiplication.
2
Solution: The set of cube roots of unity is G = {1, , }. Let us form the composition table as given
below:

I 2
1 1 2
2 3 = 1
2 2 3 = 1 4 =

(G ) Closure Axiom: Since each element obtained in the table is a unique element of the given
1
set G, multiplication is a binary operation. Thus the closure axiom is satisfied.
(G ) Associative Axiom: The elements of G are all complex numbers and we know that
2
multiplication of complex number is always associative. Hence associative axiom is also
satisfied.
(G ) Identity Axiom: Since row 1 of the table is identical with the top border row of elements
3
of the set, 1 (the element to the extreme left of this row) is the identity element in G.
(G ) Inverse Axiom: The inverse of 1, , 2 are 1, 2 and respectively.
4
(G ) Commutative Axiom: Multiplication is commutative in G because the elements equidistant
5
with the main diagonal are equal to each other.
The number of elements in G is 3. Hence (G,.) is a finite group of order 3.

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