Which one of the following is not necessarily a property of a group?

Correct answer: Commutativity is not necessary for a group.

A group must satisfy these properties:

• Closure: the operation on any two elements produces another element of the set.

• Associativity: (a * b) * c = a * (b * c) for all elements a, b, c.

• Identity: there exists an element e such that e * a = a * e = a for every a.

• Inverse: for each element a there exists b with a * b = b * a = identity.

Explanation:

Commutativity means a * b = b * a for all elements a and b. This property is required only for Abelian (commutative) groups, not for all groups.

Example: The symmetric group S3 (the permutations of three objects) satisfies closure, associativity, identity, and inverses, so it is a group, but it is not commutative because some permutations do not commute. Thus commutativity is not a necessary condition for being a group.