Which one of the following in NOT necessarily a property of a Group?

Answer: Commutativity is not necessarily a property of a group.

Reason: By definition, a group must satisfy the following axioms:

• Associativity: the operation is associative.

• Identity: there exists an identity element e with ea = ae = a for all a.

• Inverses: for every element a there exists an inverse b with ab = ba = e.

Commutativity (ab = ba for all a, b) is not part of the group axioms. Groups that do satisfy commutativity are called abelian, but many groups are non-abelian.

• Example: The symmetric group S3 (permutations of three objects) is a group but not commutative. For instance, the permutations that swap elements 1 and 2 and that swap 2 and 3 do not commute.

Therefore, among the listed properties, commutativity is the one that is not necessarily true for every group.