Which of the following statements is FALSE? A. The set of rational numbers is an abelian group under addition. B. The set of integers is an abelian group under addition. C. The set of rational numbers forms an abelian group under multiplication. D. The set of real numbers excluding zero is an abelian group under multiplication.

GATE 1996 · Discrete Mathematics · Group Theory · medium

Answer: C. The set of rational numbers forms an abelian group under multiplication

1. Verify the additive groups: (Q, +) and (Z, +) both have identity 0 and additive inverses, so A and B are true.
2. Test (Q, *) for inverses: 0 is a rational number with no multiplicative inverse, so (Q, *) is not a group.
3. Confirm (R\{0}, *) is fine: Removing 0 leaves every element invertible, so D is a true abelian group.