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