The set of Integers (Z) with multiplication:

Satisfies the other conditions for a group, but does not have an identity element.

Satisfies the other conditions for a group, but does not have inverses.

Satisfies the other conditions for a group, but does not satisfy the associative law.

Satisfies the other conditions for a group, but is not closed under the operation.

Option B

If a ∈ Z, a × 1 = a is always true, so 1 is the identity element.

• If a ∈ Z, a × (1/a) = 1 and (1/a) × a = 1 (the identity element), so a and 1/a are inverses of each other.

However 1/a is not generally an integer, so most integers do not have inverses under multiplication.

• For any three integers a, b and c in Z: a × (b × c) = (a × b) × c

⇒ The associative law has been satisfied.

• If a and b are any two integers, (a × b) is also an integer

⇒ The closure law has been satisfied.

Therefore, three conditions for a group are satisfied by the set Z under multiplication, but the inverse condition is not satisfied.

- 2024-06-12 03:15
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- 25 Full Marks
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