Quiz 4-6_Abstract

What is the unique element in a group that leaves other elements unchanged when combined with them?

• Inverse element

• Commutative element

• Associative element

• Identity element

In a group, for any elements (a, b, c) in (G), which property holds: (ab)c = a(bc)?

• Commutativity

• Associativity

• Distributivity

• Inverseness

In a group, if the operation is commutative, which property holds true?

• (ab)c = a(bc)

• a(bc) = (ab)

• None of the above

• ab = ba

Which condition is NOT necessary for a set to be considered a group?

• Closure under the operation

• Existence of an identity element

• Existence of an inverse element for each element

• Commutativity of the operation

In a group, if every element has an inverse, the group is said to be:

• Commutative

• Associative

• Abelian

• Inversely closed

Which statement is true about the identity element in a group?

• It is unique

• It is the same as the inverse element

• It is not necessary for the group to have an identity element

• It only exists in non-abelian groups

Which property states that for any element (a) in a group (G), there exists an element (b) in (G) such that ab = ba = e, where (e) is the identity element?

• Closure

• Associativity

• Inverse property

• Commutativity

If every element of a group (G) commutes with every other element of (G), the group is called:

• Abelian

• Non-associative

• Non-commutative

• Idempotent

Which property ensures that for any elements (a, b) in a group (G), (ab) also belongs to (G)?

• Associativity

• Closure

• Commutativity

• Inverse property

If (G) is a group and (a, b) are elements of (G), which property ensures that (ab = e) implies (ba = e)?

• Associativity

• Closure

• Inverse property

• Commutativity