Which one of the following is NOT necessarily a property of a Group? A. Commutativity B. Associativity C. Existence of inverse for every element D. Existence of identity

GATE 2009 · Discrete Mathematics · Group Theory · medium

Answer: A. Commutativity

1. State the four group axioms: Comparing with the options: B (Associativity) = Axiom 2 — required. C (Inverse) = Axiom 4 — required. D (Identity) = Axiom 3 — required. A (Commutativity) = NOT one of the four axioms.
2. Give a non-Abelian group as evidence: The group GL(2,R) of 2x2 invertible real matrices under multiplication satisfies all group axioms but is not commutative: for example, [[1,1],[0,1]] * [[1,0],[1,1]] != [[1,0],[1,1]] * [[1,1],[0,1]]. So commutativity is NOT necessarily a property of every group.