Page 31 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Notes
Example 9: Prove that the set {1, –1, i, –i} is abelian multiplicative finite group of order 4.
Solution: Let G = {1, –1, i, –i}. The following will be the composition table for (G,.)

1 –1 i –i
1 1 –1 i –i
–1 –i 1 –i –i
i i –i –1 1
–i –i i 1 –1

(G ) Closure Axiom: Since all the entries in the composition table are elements of the set G, the
1
set G is closed under the operation multiplication. Hence closure axiom is satisfied.
(G ) Associative Axiom: Multiplication for complex numbers is always associative.
2
(G ) Identity Axiom: Row 1 of the table is identical with that at the top border, hence the
3
element 1 in the extreme left column heading row 1 is the identity element.
(G ) Inverse Axiom: Inverse of 1 is 1. Inverse of –1 is –1. Inverse of i is –i and of –i is i. Hence
4
inverse axiom is satisfied in G.
(G ) Commutative Axiom: Since in the table the 1st row is identical with 1st column, 2nd row
5
is identical with the 2nd column, 3rd row is identical with the 3rd column and 4th row is
identical with the 4th column, hence the multiplication in G is commutative.
The number of elements in G is 4. Hence G is an abelian finite group of order 4 with respect to
multiplication.

General Properties of Groups

Theorem 1: The identity element of a group is unique.
Proof: Let us suppose e and e are two identity elements of group G, with respect to operation o.
Then e o e = e if e is identity.

and e o e = e if e is identity.
But e o e is unique element of G, therefore,
e o e = e and e o e = e e = e
Hence the identity element in a group is unique.

Theorem 2: The inverse of each element of a group is unique, i.e., in a group G with operation o
–1
–1
–1
for every a G, there is only one element a such that a oa = a o a = e, e being the identity.
Proof: Let a be any element of a group G and let e be the identity element. Suppose there exist
–1
a and a two inverses of a in G then
–1
a o a = e = a o a –1
and a o a = e = a o a
Now, we have
–1
–1
a o (a o a ) = a o e (since a o a = e)

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