The set {1, 2, 3, 5, 7, 8, 9} under multiplication modulo 10 is not a group. Given below are four possible reasons. Which one of them is FALSE? A. It is not closed B. 2 does not have an inverse C. 3 does not have an inverse D. 8 does not have an inverse

GATE 2006 · Discrete Mathematics · Group Theory · medium

Answer: C. 3 does not have an inverse

1. Check which elements have inverses mod 10: gcd(2,10) = 2, so 2 has no inverse (B is TRUE). gcd(3,10) = 1, so 3 DOES have an inverse: 3*7=21≡1 (mod 10), and 7 is in S. So C is FALSE. gcd(8,10) = 2, so 8 has no inverse (D is TRUE).
2. Verify closure failure (option A): 2 * 5 = 10 ≡ 0 (mod 10). Since 0 is not in S, the set is not closed. So A is TRUE.