# Examples of Group

__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 an 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 the 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 to the 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, the closure axiom is satisfied.

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

for all

Hence, the associative axiom is satisfied.

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

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

so that is the multiplicative inverse of . Thus the 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 the 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.