__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. |