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

A group is a set G together with a binary operation that must satisfy four axioms:

• Closure — combining any two elements of G with the operation gives a result that is also in G.

• Associativity — (ab)c = a(bc) for all a, b, c in G.

• Existence of identity — there is an element e with ea = ae = a for every a in G.

• Existence of inverse for every element — for each a in G there is some b in G with ab = ba = e.

These four — closure, associativity, identity, and inverse — are the complete list of group axioms; nothing more is required.

Three of the four listed options name three of these axioms directly — Associativity, Existence of identity, and Existence of inverse for every element — so a structure missing any one of them is not a group at all. Commutativity (ab = ba for every pair of elements a, b) does not appear anywhere in that axiom list.

This is confirmed by example: the symmetric group S3, the set of all 6 permutations of 3 objects under composition, satisfies closure and associativity, has an identity (the do-nothing permutation), and every permutation has an inverse — so S3 is a group. Yet S3 is non-abelian: swapping the first two objects and then swapping the last two objects gives a different overall permutation than performing those two swaps in the reverse order. A non-commutative structure like S3 still qualifies fully as a group; a group that happens to be commutative is given the extra name abelian, showing commutativity is optional, not required.

Therefore, commutativity is the property that is NOT necessarily true of a group: associativity, identity, and inverse are required, but commutativity is not.