__Example__ 1:

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.

**(G1) Closure Axiom.** We know that the sum of any two integers is also an integer, i.e., for all, . Thus is closed with respect to addition.

**(G2) Associative Axiom .** Since the addition of integers is associative, the associative axiom is satisfied, i.e., for Such that

**(G3) Existence of Identity.** We know that is the additive identity and, i.e.,

Hence, additive identity exists.

**(G4) Existence of Inverse. **If , then . Also,

Thus, every integer possesses additive inverse. Therefore is a group with respect to addition.

Since addition of integers is a commutative operation, therefore

Hence is an Abelian group. Also, contains an infinite number of elements.

Therefore is an Abelian group of infinite order.

__Example__ 2:

Show that the set of all non-zero rational numbers with respect operation of multiplication is a group.

__Solution__:

Let the given set be denoted by . Then by group axioms, we have

**(G1)** We know that the product of two non-zero rational numbers is also a non-zero rational number. Therefore is closed with respect to multiplication. Hence, closure axiom is satisfied.

**(G2)** We know for rational numbers.

for all

Hence, associative axiom is satisfied.

**(G3) **Since, the multiplicative identity is a rational number hence identity axiom is satisfied.

**(G4)** If , then obviously, . Also

so that is the multiplicative inverse of . Thus inverse axiom is also satisfied. Hence is a group with respect to multiplication.

__Example__ 3:

Show that , the set of all non-zero complex numbers is a multiplicative group.

__Solution__:

Let Here is the set of all real numbers and .

**(G1) Closure Axiom.** If and, then by definition of multiplication of complex numbers

Since , for . Therefore, is closed under multiplication.

**(G2) Associative Axiom.**

for .

**(G3) Identity Axiom.** is the identity in .

**(G4) Inverse Axiom.** Let , then

Where and

Hence is a multiplicative group.