Page 27 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Note: It should be noted that the smallest group for a given composition is the set {e} consisting Notes
of the identity element e alone.

Illustrative Examples

Example 4: Show that the set of all integers ……, –4, –3, –2, –1, 0, 1, 2, 3, 4, … is an infinite
abelian group with respect to the operation of addition of integers.
Solution: Let us test all the group axioms for abelian group.

(G ) Closure Axiom: We know that the sum of any two integers is also an integer, i.e., for all
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a, b I, a + b I. Thus I is closed with respect to addition.
(G ) Associativity: Since the addition of integers is associative, the associative—axiom is
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satisfied, i.e., for a, b, c I.
a + (b + c) = (a + b) + c
(G ) Existence of Identity: We know that O is the additive identity and O I, i.e.,
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O + a = a = a + O a I
Hence additive identity exists.
(G ) Existence of Inverse: If a I, then –a I. Also,
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(–a) + a = O = a + (–a)

Thus every integer possesses additive inverse.
Therefore I is a group with respect to addition.
Since addition of integers is a commutative operation, therefore a + b = b + a a, b I.
Hence (I, +) is an abelian group. Also, I contains an infinite number of elements. Therefore (I, +) is
an abelian group of infinite order.

Example 5: Show that the set of all even integers (including zero) with additive property
is an abelian group.

Solution: The set of all even integers (including zero) is
I = {0, ± 2, ± 4, ± 6…}
Now, we will discuss the group axioms one by one:
(G ) The sum of two even integers is always an even integer, therefore closure axiom is satisfied.
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(G ) The addition is associative for even integers, hence associative axiom is satisfied.
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(G ) O I, which is an additive identity in I, hence identity axiom is satisfied.
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(G ) Inverse of an even integer a is the even integer –a in the set, so axiom of inverse is satisfied.
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(G ) Commutative law is also satisfied for addition of even integers. Hence the set forms an
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abelian group.

Example 6: Show that the set of all non-zero rational numbers with respect to binary
operation of multiplication is a group.

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