To prove that a binary operation on a certain set, say makes it a group, we need to show that four properties are satisfied. Let's illustrate this by examples.
Problem 1
Let be a set of non-zero rational numbers such that , . Show that is a group.
(i) Closure: If and are non-zeros, then is non-zero rational number, and so
(ii) Associative: For
Similarly,
(iii) Identity element: The number two acts as the multiplicative identity, so,
(iv) Inverse: If is non-zero, then is a no-zero rational number and serves as the multiplicative inverse of , since
Problem 2
Let , i.e., the set of all real numbers greater than . Define
for Show that the operation makes a group.
Solution
For to be a group, four properties: closure, associative, identity element and inverse must be satisfied.
(i) Closure: If , then so and therefore multiplying by both sides, it follows immediately that
Hence the operation is closed on .
(ii) Associative: For we have
On the other hand, we have
Thus
(iii) Identity: Hence, 2 is the identity element for the operation .
(iv) Inverse: Given any we need to solve the equation This gives the equation
This solution belongs to since
NB: You can also check that
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Properties of a Group - Solved Examples
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