Show that set of integers Z is not a group under the binary operation multiplication. Justify each of the group properties that holds or not. | Query Point Official

Why the Set of Integers Z is Not a Group Under Multiplication

The set of integers Z is considered under the binary operation of multiplication. We will check each group property to see if it holds.

Group Properties Check

Group Property	Check	Conclusion
Closure	For any integers a, b ∈ Z, a × b ∈ Z.	✅ Holds. Multiplying two integers always gives an integer.
Associativity	(a × b) × c = a × (b × c) for all a, b, c ∈ Z.	✅ Holds. Multiplication of integers is associative.
Identity Element	There exists e ∈ Z such that a × e = e × a = a for all a ∈ Z.	✅ Holds. The identity element is 1.
Inverse Element	For each a ∈ Z, there exists a⁻¹ ∈ Z such that a × a⁻¹ = 1.	❌ Fails. Only 1 and -1 have integer inverses. Other integers (e.g., 2) have inverses (1/2) which are not integers.

Conclusion

The set of integers Z is not a group under multiplication because it fails the inverse element property. While it satisfies closure, associativity, and has an identity, most integers do not have multiplicative inverses in Z.

FAQ

Q1: Can Z form a group under addition?

Yes. Under addition, integers form a group because every integer has an additive inverse in Z.

Q2: What is a group?

A group is a set with a binary operation that satisfies closure, associativity, has an identity element, and each element has an inverse.

Q3: Can a subset of Z form a group under multiplication?

Yes. For example, the set {1, -1} forms a group under multiplication.

Related Topics

For more biology definitions, visit Group Theory Notes & MCQs.

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